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IMaps, plates, charts, etc.. may be filmed at different reduction ratios. Those too large to be entirely included in one exposure ara filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartas, planches, tableaux, etc.. peuvent Atre filmte * des taux de r6duction difftrents. Lorsque le document est trop grand pour Atre reproduit nn un saui clichA. il est film* A partir da Tangle supArieur gauche, de gauche A droite, et de haut en bas, an prenant le nombre d'images nAcessaire. Las diagrammes suivants illustrant ia mAthode. 1 2 3 32X 1 2 3 4 5 6 1 < eDi' BOWSER'S MATHEMATICS. ACADEMIC ALGEBRA, WITH MMKKors KXAMI'LES. COLLEGE ALGEBRA, WITH XlMKIfors HXAMI'LKS. AN ELEMENTARY TREATISE ON ANALYTIC GEOMETRY, KMI'.K \('lN(i I'LANK (iKOMKTKY, AM) AN IXTHODrcTlON T(» (iEOMKTHY OF THHKK DniEN'SIONS. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, WITH NIMEKOIS EXAMPLES. AN ELEMENTARY TREATISE ON ANALYTIC MECHANICS, WITH NlMEh'ors KXAMPLICS. AN ELEMENTARY TREATISE ON HYDRO-MECHANICS, WITH MMEUors EXAMI'I.KS. if \ ifUifUUWWiniiuifMVMI ^ ^i V *^* I aCULTY of APPLIi-D SCIilNC- a Univcir-sity of "^ 'r;;-:;:o. J Elementary Treatise ON HYDROMECHANICS, t « With Numerous Examples, BY EDWARD A. BOWSER, LL. D., PROFESSOR OF MATHEMATICS AND ENGINEERING IN RUTGERS COLLEGE. THTRD EDITTON, \ NEV D/ TAN NOST NE\A/^ YORK :J7 RAND-tJOMPANY 23 Murray and 27 Warren Sts, \ 1889. ' S ^^'1 Copyright, 1885, By E. A. BOWSER. / ELICntOTYPID BY SMITH & McDnuOAL, 82 BlIKHAN St., N.Y. PREFACE. THE present work on Hydromechanics is designed as a text-book for Scientific Schools and Colleges, and is prepared on the same general plan as the author's Analytic Mechanics, which it is intended to follow. Like the Ana- lytic Mechanics, it involves the use of Analytic Geometry and the Calculus, though a geometric proof has been intro- duced wherever it seemed preferable. The book is divided into two parts, namely, Hydrostatics and Hydrokinetics. The former is subdivided into three, and the latter into four chapters ; and at the ends of the chapters a large number of examples is given, with a view to illustrate every part of the subject. Many of these ex- amples were prepared specially for this work, and are prac- tical questions in hydraulics, etc., taken from e very-day life. In writing this treatise, the aim has been to enunciate clearly the fundamental principles of the theory of Hydro- mechanics, to explain some of the most important applica- tions of these principles, and to render more general the study of this interesting science, by presenting as simple a view of its principles as is consistent with scientific accu- racy. Throughout the work a careful distinction has been made between those propositions which are necessarily true, being deduced from the definitions and axioms of the sub- ject, and those resiilt? which are empirical, IV PREFACE. \\\ ill! element'iry work of this kind there is not room for niiitli lluit is new. I liave drawn freely upon the writings of many of the best autliors. The works to whicli I urn l)riiicii ally indebted, and which are here named for con- venience of reference by the student, are those of Besani, Lamb, Kankine. Boucharlat, Weisbach, Cotterill, Bland, Jiiniieson, Fanning, Pratt, Renwick, Stanley, Tate, Desch:;- nt'l, Bossut, d'Aubuisson, Poucelot, Eytelvvein, Prony, Starrow, Goodeve, Galbraith, Gregoi-y, Twisden, Bartlett, AYood, Smith, Olmsted, Morin, Humphreys and Abbot, Fiiirbairn, Colyer, Barrow, and the Encyclopaedia Britan- nica. • My thanks are again due to my friend and former pupil, Mr. R. W. Prentiss, of the Nautical Almanac Office, and formerly Fellow in Mathematics at the Johns Hopkins University, for reading the MS. and for valuable sugges- tions. E. A. B. RUTOEKS COLLEOE, New Bkunswick, N. J., April, 1885. \\ i TABLE OF CONTENTS --•♦■♦■•♦- PART I. HYDROSTATICS. CHAPTER I. EQUILIBRIUM AND PRESSURE OF FLUIDS. ABT. TAOB 1. Definivions — Hydrostatics, Hydrokinetics 1 2. Three States of Matter 1 3. A Perfect Fluid 3 4. Direction of Pressure 3 5. Solidifying a Fluid 4 6. Measure of the Pressure of Fluids 4 7. Pressure the Same in Every Direction 3 8. Equal Transmission of Fluid Pressure 6 9. Equilibrium of Pressures 8 10. Pressure of a Licjuid at any Depth 10 11. Free Surface of a Liquid at Rest 13 12. Common Surface of Two Fluids 15 13. Two Fluids in a Bent Tube 16 14. Pressure on Planes 17 15. The Whole Pressure 18 16. Centre of Pressure 21 17. Embankments 27 18. Embankment when the Face on the Water Side is Vertical. . . 27 19. Embankment when the Fnce on the Water Side is Slantiiifr . . 29 20. Pressure upon Both Sides of a Surface 33 21. Rotating Liquid 35 22. Pressure at any Point of a Rotating Liquid 37 23. Strength of Pipes and Boilers 39 Examples 48 CONTENTS. CHAPTER II. EQUILIBRIUM OF FLOATING BODIES— SPECIFIC GRAVITY. ART. '*«" 24. Upward Pressure, Buoyant Effort 50 25. Conditions of Equilibrium of an Immersed Solid 52 26. Depth of Flotation 54 27. Stability of Equilibrium 57 28. Position of the Metacentre ; Measure of Stability 60 29. Specific Gravity 66 30. The Standard Temperature 67 31. Methods of Finding Specific Gravity 69 32. Specific Gravity of a Solid Broken into Fragments 72 33. Specific Gravity of Air 73 84. Specific Gravity of a Mixture 78 35. Weights of the Components of a Mechanical Mixture 75 36. The Hydrostatic Balance 76 37. The Common Hydrometer 77 38. Sikes's Hydrometer 78 89. Nicholson's Hydrometer 79 Examples 80 CHAPTER III. EQUILIBRIUM AND PRESSURE OF GASES — ELASTIC FLUIDS. 40. Elasticity of Gases 88 41. Pressure of the Atmosphere 89 42. Weight of the Air 90 43. The Barometer 91 44. The Mean Barometric Height 92 45. The Water-Barometer 93 46. Manometers 93 47. The Atmospheric Pressure on a Square Inch 94 48. Boyle and Mariotte's Law 95 49. Effect of Heat on Gases 98 50. Thermometers— Fahrenheit, Centigrade, Reaumur 99 51. Comparison of the Scales of these Thermometers 100 52. Expansion of Mercury 101 53. Dalton's and Gay-Lussac's Law 101 54. Pressure, Temperature, and Density 103 55. Absolute Temperature 105 CONTENTS. Yii ABT. PAGl 56. The Pressure of a Mixture of Gases 106 57. Mixture of Equal Volumes of Gases 107 58. Mixture of Unequal Volumes of Gases 108 59. Vapors, Gases 108 60. Formation of Vapor, Saturation 109 61. Volume of Atmospheric Air without its Vapor 110 62. Change of Volume and Temperature 110 03. Formation of Dew— the Dew Point 112 64. Pressure of Vapor in the Air 112 65. Effect of Compression or Dilatation on Temperature 118 66. Expansion of Bodies — Maximum Density of Water 118 67. Thermal Capacity— Unit of Heat— Specific Heat 115 68. Specific Heat at a Constant Pressure, and at a Constant Volume 116 69. Sudden Compression of a Mass of Air 118 70. Mass of the Earth's Atmosphere 120 71. The Height of the Homogeneous Atmosphere 120 72. Necessary Limit to the Height of the Atmosphere 121 78. Decrease of Density of the Atmosphere 122 74. Heights Determined by the Barometer 124 Table of Specific Gravities 130 Examples 181 n ♦ «» PART II. HYDROKINETICS. CHAPTER I. MOTION OF LIQUIDS — EFFLUX — RESISTANCE AND WORK OF LIQUIDS. 75. Velocity of a T^iquid in Pipes 136 76. Velocity of Efflux 137 77. The Horizontal Range 140 78. Time of Discharge when the Height is Constant 141 Vlll ART. 79. 80. 81. 82. 83. 84. 85. 80. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 5)9. 100. 101. CONTENTS. PAUE Time of Emptying any Vessel 142 Time of Emptying n Cylinder into a Vacuum 144 Time of Einj)tying ii Paraboloid 145 Cylindriral Vessel with Two Small Orifices 145 Orifice; in the Side of a Conical Vessel 146 Velocity of Efflux through an Orifice in the Bott(t:ii 147 Kfctiingular Orifice in the Side of a Vessel 149 Triangular Orifice in the Side of a Vessel 151 Time of Emptying any Vessel through a Vertical Orifice 156 Efflux from a Vessel in Motion 158 Efflux from a Kotating Vessel 160 The Clepsydra, or Water-Clock 161 The Vena Contracta 162 Coefficient of Contraction 163 Coefficient of Velocity 164 Coefficient of Efflux 164 J^fflux through Sliorr Tubes, or Ajutages 165 Coefficient of Resintance 167 Uesistanco and Pressure of Fluids 170 Work and Pressure of a Stream of Water 172 Impact against any Surface of Revolution 174 Oblique Impact 178 Maximum Work done by the Impulse 180 Examples , 181 CHAPTER II. MOTION OF WATER IN PIPKS AND OPEN CHANNELS. 102. Resistance of Friction 185 103. Motion of Water in Pipes 186 104. Uniform Pipe connecting Two Reservoirs 188 105. Coefficient of Friction for Pipes 191 106. The Quantity Discharged from Pipes 194 107. The Diameter of Pipes 197 108. Sudden Enlargement of Section 199 109. Sudden Contraction of Section 201 1 10. Elbows 204 111. Bonds 200 \\\ti. Equivalent V\\)e% 307 1116. • Discharge Diminishing Uniformly 808 1 CONTENTS. ix ART. TAGE 112. General Formula for all the HosistanceH 200 113. Flow of Water in Kivers ami CaiialH 21 1 114. Ditferent Velwities in a Croys-Seotion 212 115. Transverse Section of the Stream 2ir) 116. Mean Velocity 210 11?. Ratio of Mean to Greatest Stirface Velocity 216 118. Processes for Gauging Streams 218 119. Most Economical Form of Transverse Section 221 120. Trapezoidal Section of Least Resistance 222 121 . Uniform Motion 224 122. Coefficients of Friction 226 123. Variable Motion 228 124. Bottom Velocity at which Scour Commences 232 125. Transporting Power of Water 23:> 126. Back Water 235 127. River Bends 236 Examples 237 CHAPTERIII. MOTION OF ELASTIC FLUIDS. 128. Work of the Expansion of Air 242 129. Velocity of Efflux of Air according to Mariotte's Law 244 130. Efflux of Moving Air 247 131. Coefficient of Efflux 249 132. The Quantity Discharged 250 133. Coefficient of Friction of Air 251 134 Motion of Air in Long Pipes 252 135. Law of the Expansion of Steam 254 136. Work of Expansion of Steam 256 137. Work of Steam at Efflux 257 138. Work of Steam in the Expansive Engine 259 Examples 260 CHAPTER IV. HYDROSTATIC AND HYDRAULIC MACHINES. 139. D«>finitions 263 140. The Hydrostatic Bellows 263 X CONTENTS. ART. PAOB 141. The Siphon ; 264 142. The Diving Bell 266 143. The Common Pump (Suction Pump) 268 144. Tension of the Piston-Rod 270 145. Height through which Water Rises in One Stroke 271 146. The Lifting Pump 273 147. The Forcing Pump 274 148. The Fire Engine 276 149. Bramah's Press 276 150. Hawksbec's Air-Pump 277 151. Smeaton's Air-Pump 279 152. The Hydraulic Ram 280 153. Worli of Water Wheels 282 154. Work of Overshot Wheels 283 155. Work of Breast Wheels 284 156. Work of Undershot Wheels 285 157. Work of the Ponceiet Water Wheel 286 158. The Reaction Wheel ; Barker's Mill 288 159. The Centrifugal Pump 290 160. Turbines 292 Examples 294 I HYDROMECHANICS —> ♦ >» PART I. HYDROSTATICS. CHAPTER I. EQUILIBRIUM AND PRESSURE OF FLUIDS, 1 . Definitions.— Hydromechanics is the science wTiich treats of the equilibrium and motion of fluids. It is accord- ingly divided into two parts, Hydrostatics and Hydrokinetics. Hydrostatics treats of the equilibrium of fluids. Hydrokinetics treats of the motion of fluids. The object of the science of Hydrostatics is to determine the equilibrium and pressure of fluids, the nature of the action which fluids exert upon one another and upon bodies with which they are in contact, and the weight and pressure of solids immersed in them, and to explain and classify, under general laws, the different phenomena to which they give rise. 2. Three States of Matter.— Bodies exist in three different states, depending upon the manner in which their particles are held together. They are either solid or fltiid ; and the latter are either liquid or yaseons. Solid bodies are those whose particles are held together so firmly that a certain force is necessary to change their forms fi A FEKFECT FLUID. or to produce a separation of their particles. If a solid be reduced to the finest powder, still each grain of the powder is a solid body, and its particles are held together in a de- terminate shape. Fluids are bodies, the position of wliose particles in ref- erence to one another is changed by tlie smallest force. The distinguishing property of a fluid is the perfect facil- ity with which its particles move among one another, and as a consequence its readiness to change its form under the influence of the slightest effort. Fluids are of two kinds, Uquiils and gases. In a liquid there is a perceptible cohesion among its particles ; but in a (jas the particles mutually rei)el one another. Every solid body possesses a peculiar form of its own, and a definite volume; liquids have only a definite volume, but no pecu- liar form ; and gases have neither one nor the other. If a liquid, such as water, be poured into a tumbler, it will lie at the bottom, and will be separated by a distinct surface from the air above it; but if ever so snuiU a quantity oi gas be introduced into an empty and closed vessel, it Avill im- mediately expand so as to fill tiie whole vessel, and will exert some amount of pressure upon the interior surface. 3. A Perfect Fluid — Fluids differ from each other in the degree of cohesion of their particles, and the facility with which they will yield to the action of a force. Many bodies which are met with in nature, such as water, mer- cury, air, etc., possess the properties of fluids in an eminent degree, Avhile others, such as oil, tallow, the sirups, etc., possess a less degree of fluidity. The former are called perfect fluids^ and the latter viscous or imperfect fluids. In this work, only perfect fluids will be considered. .7 prr/'eef fli id is an ag^re^aNon. of parti rlrs which yield at once to the slightest ejfort made to separate them from one another. DIUKVTIoy OF PRJCSSURE. 3 '^ Fluids are divided into two classes incompressible and compressible. The former are sometimes called inelastic and the latter elastic fluids. Incompressible fluids are those which retain the same volume under a variable pressure. Compressible fluids are those in which the volume is diminished as the pressure upon it is increased, and increased as the pressure upon it is diminished. The term incompressible cannot strictly be applied to any body in nature, all bring more or less compressible. But on account of the enormous power required to change, in any sensible degree, the volumes of liquids, they are treated in most of the researches in hydrostatics as incom- pressible or inelastic fluids. It was shown by Canton, in 1761, that water under a pressure of one atmosphere, i. e., of about one ton on each square foot of surface, undergoes a diminution of forty-four millionths of its total volume.* All liquids are therefore regarded as incompressible. Water, mercury, wine, etc., are generally ranged under this class. The f/ases are highly compressible, such as air and the dif- ferent vapors. 4. The Direction of the Pressure of a Fluid on a Surface, — If an indefinitely thin plate be made to di- vide a fluid in any direction, no resistance will be offered to the motion of the plate in the direction of its plane, /. e., there will be no tangential resistance of the nature of fric- tion, such, for instance, as would be exerted if the plate were pushed between two flat boards held close to each other. Hence the following fundamental proi)erty of a fluid is obtained from its definition : TIte pressure of a fluid is always normal to any sur- face with which it is iu contact. * Galbraith's Hydrostatics ; Oregory's Uydrostiilica. The coinpro( the j^ressure at IJ, wu have pa = waz ; or, p = wz ; (1) that is, the pressure at any depth varies as the depth below the surface. Similarly, let B and C be any two points in the same ver- tical line, and let the cylinder BC, be solidified; then, from what has just been shown, the pressures at B and must differ by the weight of the cylinder BC, i. e., the press- ure at C is greater than that at B by the weight of a column of liquid whose base is equal to the area C, and whose height is BC. Hence, if p and p' be the pressures at B and C, and BC = z, we have pa — pa waz ; or. p —p = 2oz; m that is, the difference of the pressures at any two points varies as the vertical distance between the points. Cor. 1. — If W be the weight of a mass M, of fluid, then (Anal. Mechs., Art. 34), we have W = Mg. (3) If Y be the volume of the mass M, of fluid, and p be its density, then (Anal. Mechs., Art. 11), we have M= Vp. W = gpV. (4) (6) For a unit of volume we have F — 1, therefore (5) be- comes W = gp. From (1) we have. pa = waz = W = gpV [from (5)], .' 1)1 or. t'RESStJIiS OF A LIQUID AT AXY DFJ'Tlt. pa = f/paz (since V=az)'y (6) Cor. a.— If A be the area of tlie base of a vessel, h its height, and P tlie whole pressure on the base, wo have, from (G), P = yphA. (8) That is, the pressure of a liquid on any horizontal area is equal to the weight of a column of the liquid whose base is equal to the area, and whose height is equal to the height of the surface of the liquid above the area. It is evidently immaterial whether the surface pressed is that of the base of the vessel or a horizontal surface of an immersed solid. Cor. 3. — Since the weight of a cubic foot of water = 1000 ozs. = 63.5 lbs., we have, for the pressure on the bottom of any vessel containing water, P = 62.5/^.4 lbs., (9) where h is the height in feet of the surface of the water above the base, and A the area of the base in square feet. Cor. 4. — The pressure on the base of any vessel is independent of the form of the vessel. Thus, if a hollow cone, vertex upwards, be filled with water, and if r be the radius of the base and h the he'ght of the cone, we have for the pressure on the base, P = gpirrVi [from (8)], or, P = Q2.67TrVi [from (9)] ; that is, the pressure on the base is the same as if the cone were a cylinder of liquid of the same base and height as the FltEE SURFACE OF A LIQUID AT REST. 18 cone; the pressure is three tioies the weight of the enclosed water. This increased pressure on the base is caused by the re- action of the curved surface of the cone. Tlie pressure on the curved surface consists of an assemblage of forces whose vertical components all point downwards and react upon the base. EXAM PLES. 1. If a surface of one square inch be placed in a vessel completely filled with water, and if the pressure upon it be 2 lbs., what will be the pressure on one square inch placed at a level 75 inches lower ? Here A = one square inch, A = 75 inches, and P and P' are the pressures at the upper and lower points; there- fore we have, from (3) and (8), P'-P = 252.5* X 76 = 18937.5 grains = 2.705 lbs. .-. P' = 2.705 + 2 = 4.705 lbs. 2. If the pressure on the upper surface, whose area is a circle of half an inch radius, is 1.5 lbs., find the pressure on another circular area whose radius is one inch, placed at a depth 10 feet lower in the water. Ans. 19.5986 lbs. 11. The Free Surface of a Liquid at Rest is a Horizontal Plane.— Let ABCD represent the section of a vessel containing a liquid subject to the action of gravity; then will its free surface be horizontal. For, if the free surface is not horizontal, suppose it to be the curved line, APB. Take any point P, of the surface where the tan- gent to the curve is not horizontal ; let * Hie wdght of one cubic inch of water at the standard tempentore is S6SJf gralna. u FREE SURFACE OF A LIQUID AT REST, the vertical line PO, be drawn to represent the weight of the particle of liquid at P, and resolve this weight into two components PR and PQ, the former perpendicular, and the latter parallel to the surface. The first of these is opposed by the reaction of the surface ; the second, being unopposed, causes the particle to move downwards to a lower level. It is evident, therefore, that if the free surface be one of equi- librium, it must at each point be perpendicular to the direc- tion of gravity, i. e., it must be horizontal. Cor. 1. — Since the directions of gravity, acting on parti- cles remote from each other, are convergent to the earth's centre, nearly, large surfaces of liquids are not plane, but curved, and conform to tlie general figure of the earth. But, for small areas of surface the curvature cannot be de- tected, because the deviation from a plane is infinitesimal. Cor. 3. — The pressure of the ^mosphere is found to be about 14.73 lbs. to a square incii, or very nearly 15 lbs. The pressure, therefore, on any given area can be calculated, and if tt be the atmospheric pressure on the unit of area, the pressure at a depth 2; of a h'quid, the surface of which is exposed to the pressure of the atmosphere, will be, from (7) of Art. 10, P =gpz + TT. (1) Cor. 3.— Since the pressures are equal when the depths are equal (Art. 10), it follows that the areas of equal press- ure are also areas of equal depth ; therefore, since the surface of a liquid is a horizontal plane, an area of equal pressure is everywhere at the same depth below a horizontal plane, /. e., an area of equal pressure is a horizontal plane ; and, conversely, the pressure of a liquid at rest at all points of a horizontal plane is the same. Hence it appears that when the pressure on the surface of a liquid is either zero or is equal to the constant atmos- pheric pressure, all points on its surface must be in the I UL I COMMON SURFACE OF TWO FLUIDS. 15 eight of into two and the opposed opposed, jvel. It of equi- lie direc- •n parti- ; earth's ine, but e earth, t be de- isimal. i to be 15 lbs. uilated, )f area, which e, from (1) depths 1 press- ce the f equal izontal zontal Ud at same. mrface atmos- in the same horizontal plane, even though the continuity of the surface be interrupted by the immersion of solid bodies. // any number of vessels, containing the same liquid, are in coimnunication, the liquid stands at the sam,e height in each vessel. This sometimes appears under the form of the assertion that liquids m^aintain their level. Rem. — The construction by which towns are supplied with water furnishes a practical illustration of this princi- ple. Pipes, leading from a reservoir placed on a height, carry the water, underground or over roads, to the tops of houses or to any point provided that no portion of a pipe is higher than the surface of the water in the reservoir. 12. The Common Surface of Two Fluids.— Let AD be the upper surface of the lighter fluid, and BO the com- mon surface of the two fluids; AD is hori- zontal (Art. 11). Let P and Q be two points in the heavier liquid, both equally distant from the surface AD, and therefore in the same horizontal plane. Draw the vertical lines Pa and Qfi, meeting the common sur- face of the fluids in c and d. Let w be the weight of a unit of volume of the upper fluid, and w' that of the lower. Then we have / ■ % > 9 -£?- g^ 3 B 8 K C Fig. 7 >■:»' and * pressure at P = 2v''cF -\- to-ac; pressure at Q = w''dQ -\- W'bd. Since the pressures at P and Q are equal (Art. 11, Cor. 3), they being in the same horizontal plane, we have But w'-cl* -f W'Oc = w''dQ + W'bd. cF + ac = dQ -{- bd (1) 16 TWO FLUIDS IN A BENT TUBE. multiplying (3) by tv, and subtracting the result from (1), wo have , , . ^,^ {w' -w)cF = {w' - w) (1(1, .'. cF = dQ, and hence BC is horizontal. That is, the common surface of two fluids that do not mix is a horizontal plane. Cor.— This proposition is- true, whatever be the number of fluids; the common surfaces are all horizontal. If, therefore, the number be infinite, or the density of the fluid vary according to any law, the surface of each will still be horizontal.* • 13. Two Fluids in a Bent Tube.— Let A and C be the two surfaces, B the common surface, @C and p, p' the densities of AR and BC. Let z and z' represent the heights of the surfaces A and C, above the com- mon surface B, and take B' in the denser fluid in the same horizontal B- plane as B. Then we have, Fig. 8 the pressure at B = gpz [(7) of Art. 10] ; the pressure at B' = gp'z', and these are equal (Art. 11, Cor. 3). ' • .-. gpz = gp'z'y Hence, when two fluids that do not mix together meet in a bent tube, the heights of their upper su?'- * See Besant*8 HydroBtatice, p. 31 ; also Bland'B Hydroetatics, p. 90. PRESSURE ON PLANES, 17 •yer sur- faces above their common surface are inversely pro- portional to their densities* 14. Pressure on Planes.— To find the pressure on a plane area in the form of a rectangle when it is just immersed in a liquid, with one edge in the surface, and its plane inclined at an angle to the vertical. Let ABCD be a vertical section perpendicular to the plane of the rectangle ; then AB is the section of the surface of the liquid, and AC (= a) is the section of the rectangle, the up- per edge b, of the rectangle being in the surface of the liquid per- pendicular to AC at A. Pass a vertical plane BC, til rough the lower edge of the rectangle, and suppose the fluid in ABC to become solid. The weight of this solid is sup- ported by the plane AC, since the pressure on BC is horizontal (Art. 4). Let R be the normal pressure on the plane AC ; resolving E horizontally and vertically, we have, for vertical forces, Rsintf = weight of ABC = ^p4AB.BC.& [(5) of Art. 10] = jigpa^ sin cos 0. R =z gpal'^a cos 0) (1) that is, the pressure on the rectangle is equal to the weight of a column of fluid whose base is the rec- * The common barometer may bo considered as an exan'ple of this principle. The air and mercury are the two fluids, If the atmosphere had the same .lensity throughout as at the surface of the earth, its height could be determined. For height of mercury in barometer : height of air :: density of air : density of mer- cury. As mercury is 10784 times as dense a air, the height of the atmosphere would be 10784 x 30 inches, or nearly 5 miles. . ! 18 TEE WHOLE PRESSURE. tangle, and whose height is equal to the depth of the middle point of the rectangle below the sur- face. Cor.— When = 0, (1) becomes R = gpdb-\a (2) = gp (area BC) (depth of middle of BC), which is the pressure on the vertical plane BC ; hence the law is the same as for the inclined plane AC. 15. The Whole Pressure.— T/^e whole pressure of a fluid on any surface with which it is in contact is the sum of the normal pressures on each of its elements. If the surface is a plane, the pressure at every point is in the same direction, and the whole pressure is the same as the resultant pressure. If it is a curved surface, the whole pressure is the arithmetic sum of all the pressures acting in various directions over the surface. The follow- ing proposition is general, and applies to curved or plane surfaces, for unit area. Let 8 be the surface, and p the pressure at a point of an element d8, of the surface. Then pdS = the pressure on the element ; (1) and since the pressure is the same in every direction (Art. 7), p will be the normal pressure on this element, whatever be its position or inclination. Hence, J J P^^ = ^he whole normal pressure, (8) the integration extending over the whole of the surface considered. THE WHOLE PRESSURE, 19 (2) (1) If gravity be the only force acting on the fluid,* we have, from (7) of Art. 10, P = gp^iy (3) z being measured vertically and positive downwards from the surface of the liquid. From (3) and (3) we have. ffpdS^ffgpzdS. (4) Calling « the depth of the centre of gravity of the surface 8, below the surface of the liquid, we have [Anal. Mechs., Art. 84, (1), p and h being constant]. which in (4) gives, "^.8 = ff^d8y f Jpd8 = gpz8. (5) for the whole pressure on the surface 8, That is, the whole pressure of a liquid on any surface is equal to the weight of a cylindrical column of the liquid whose base is a plane area equal to the area of the surface and whose height is equal to the depth of the centre of gravity of the surface below the sur- face of the liquid. Rem. — The student will now be able to appreciate more clearly the nature of fluid pressures, and to see that the action of a fluid does not depend upon its quantity, but upon the position and arrangement of its continuous por- tions. It must be borne in mind that the surface of an incompressible fluid or liquid is always the horizontal plane drawn through the highest point or points of the fluid, and that the pressure on any area depends only on its depth below that horizontal plane (Art. 10). For instance, in the construction of dock-gates, or canal-locks, it is not the * The fluid being a iiomogeneoas liquid. 20 EXAMPLES. expanse of sea outside which will affect the pressure, but the height of the surface of the sea. EXAM PLES. 1. If a cubical vessel be filled with a liquid, find the ratio of the pressures against the bottom and one of its sides. The area of the surface pressed, in each case, is the same, but the depth of the centre of gravity of the bottom is twice that of the centre of gravity of the side ; therefore the ratio is 2 : 1. 2. Find the pressure on the internal surface of a sphere when filled with water. Let a= the radius of the sphere; then the area of the surface = Aitxa^j and the deptvi of the centre of gravity of the surface below the surface of the water = a ; therefore, calling the pressure P, we have, from (5), P = gpa'^na^ = ^gpna^, which is three times the weight of the water. 3. A rectangle is immersed with two opposite sides hori- zontal, the upper one at a depth c, and its plane inclined at an angle to the horizontal. Find the a\ 'e pressure on the plane. [Let a be the horizontal side, and b the other side.] Ans, Pressure = gpab (c + h sin ^)« 4. If a cubical vessel is filled with water, and each edge of the vessel is 10 ft., find the pressure on the bottom and on a side, a cubic foot of water weighing 62^ lbs. . ( Pressure on bottom := 62500 lbs. Ans. < Pressure on side 31250 lbs. 5. A rectangular surface, 10 ft. by 5 ft, is immersed in water with its short sides horizontal, the upper side being CENTRE OF PRESSURE. %\ I 20 ft. and the lower 26 ft. below the surface of the water. Find the pressure it sustains. Ans. 32 tons.'" 6. A cylinder, closed at both ends, is immersed in a liquid so that its axis is inclined at an angle 0, to the hori- zon, and the highest point of the cylinder just touches the surface of the liquid. Find the whole pressure on the cyl- inder, including its plane ends. [Let r = the radius of the base and h = the length of the cylinder.] Ans. gpnr {k -\- r) (h sin + 2r cos 0). 7. A hemispherical cup is filled with water, and placed with its base vertical. Find the pressures on the curved and plane surfaces. A^is •I Pressure on the curved surface = Zgpna^, Pressure on the plane surface = gpna^. This example shows the distinction between the total pressure of a fiuid on a curved surface, and on that portion of it which is perpendicular to any given plane. The press- ure on the vertical plane side of the hemispherical cup might be obtained by finding the sum of the horizontal components of the actual pressures on all the elements of the curved surface. This latter pressure, called the result- ant horizontal pressure of the liquid on the surface, is equal to the pressure of the liquid on the plane base, other- wise the cup would have a tendency to move in a horizontal direction. 16. Centre of Pressure.— T/te centre of pressure of a plane area immersed in a fluid is the point of action of the resultant fluid pressure upon the plane area. It is therefore that point in an immersed plane surface or side of a vessel containing a fluid, to which, if a force equal and opposite to the rer 'Itant of all the press- * One ton = 2M0 lbs. 22 CENTRE OF PRESSURE. ii , ! '1; \ iires upon it be applied, this ft»rce would keep the surface at rest. In the case of a liquid, it is clear that the centre of press- ure of a horizontal area, the pressure on every point of which is the same, is its centre of gravity ; and since the pressure varies as the depth (Art. 10), the centre of pressure of any plane area, not horizontal, is below its centre of gravity. Let ABCD be any immersed plane area ; take the rectangular axes OX and OY, in the plane of the area. Let (.r, y) be any point P, of the area referred to these axes, and p the pressure at this point, and let EH be the line of intersection of the plane with the surface of the fluid. Fig. lo Then the pressure on the element of area = pdxdy; .'. the resultant pressure = / J pdxdy. Let Q:,y) be the ceutre of pressure; then the moment of the resultant pressure about OY = ^J J Pdxdy-y and the sum of the moments of the pressures on all the ele- ments of area about OY = // px dx dy. Therefore, since the moment of the resultant pressure is equal to the sum of the moments of the component press- ures (AnaL Mecbs., Art. 59), we have CENTRE OF PRESSURE. 23 X = and, similarly, I I pxdxdy fjP dx dy J fpy dx dy J Jpdxdy (1) y = (2) the integration extending over the whole of the area con- sidered. If polar co-ordinates be used, a similar process will give the equation Sj / / pi^ cos 6 dr do ' '''^ -. (3) z = f fpr dr do ffpr^ sin e dr dO ffpr dr dS w If the fluid be homogeneous and incompressible, and if gravity be the only force acting on it, we have [Art. 10, (7)]. p =gph, where h (= PK) is the depth of the point P below the sur- face of t 3 fluid, K being the projection of P on this sur- face, and hll being perpendicular to EH. Substituting this value oip in (1) and (2), we get X = / / hx dx dy I I hdx dy (5) M CENTRE OF PRESSURE. ffhy dx dy y = -p-ji > » I I hdxdy (6) If we take for the axis of y the line of intersection EH, of the plane with the surface of the fluid, and denote the inclination of the plane to the horizon by 0, we have PK = PM sin PMK, or, h = X sin B ; which in (5) and (6) give us, I 1 7?dx dy X = y = I j xdxdy I I xy dx dy j j xdx dy it) (8) Cor. 1. — If the axis of x be taken so that it will be sym- metrical with respect to the immersed plane, the pressures on opposite sides of this axis will obviously be equal, and the centre of pressure will be on this axis, or y = 0. Cor. 2.— Since (7) and (8) are independent of Q it ap- pears that the centre of pressure is independent of the incli- nation of the plane to the horizon, so that if p plane area be immersed ii a fluid, and then turned about its line of inter- section with the surface of the fluid as a fixed axis, the centre of pressure will remain unchanged. Rem.— The position of the centre of pressure is of great importance in practical problems. It is often necessary to know the exact effect of the pressure exerted by fluids against the sides of vessels and obstacles exposed to their EXAMPLES. M (6) the action, in order to adjust the dimensions of the latter, so that they may be strong enough to resist this pressure. Examples are furnished us in the construction of reservoirs, in which large quantities of water are collected and retained for purposes of irrigation, the supply of cities and towns, or to drive machinery, and of dykes to protect low districts from being inundated by seas and lakes and rivers in times of freshets. EXAMPLES. 1. Find the centre of pressure of a rectangle vertically immersed, and having one side parallel to the surface of the fluid, and at a given distance below it. Let a and h be the distances of the bottom and top of the rectangle from the surface of the fluid, and d the width ; take the intersection of the plane of the rectangle with the ourface of the fluid for the axis of y, and the middle point of this side for the origin, the axis of x bisecting the rectan- gle. Then from (7) we have. iB = - d ^ clx dy £ a^dx pa p\d pa I I xdxdy I xdx __ 2 a^ — ¥ Cor. — If the upper side of the rectangle is in the surface of the fluid, J = 0, and therefore we have X = fa, or the centre of pressure of a vertical rectangle, one side being in the surface of the fluid, is two-thirds the height of the rectangle belcv the surface of the fluid. The value of y is evidently zero. EXAMPLES. 2. Find tlio centre of pressure of an isosceles triangle whose base is horizontal and opposite vertex in the surface of the fluid. Let a be the altitude of the triangle and b its base. Take the intersection of the plane of the triangle with the surface of the fluid for the axis of y and the vertex for the origin, the axis of x bisecting the triangle. Then from (7) we have, b X = pa pia pa I j x'^dxdy I sfidx pa /»2o* / a^dx I / xdxdy "^o = \a. 3. A quadrant of a circle is just immersed vertically in a fluid, with one edge in the surface, ^'nd its centre of pressure. Take the edge in the surface for the axis of y, and the vertical edge for the axis of x, and let a be the radius. Then, from (7) and (8), we have pa p Va'—x* na . J J x^dxdy J x^{a^ — a^)* dx X = pa p Va'— or < y^[2(«. ■hf + ^b{2a-b)'] w Wl Coil. — If the embankment is rectangular, b = a, and (5) becomes A2 = 3a2 — . (6) C If the embankment is triangular, b = 0, and (5) becomes 7*2 = 2«2~. 19. Embankment when the Face on the Water Side is Slanting. — Find the stability of an embankment whose section is a trapezoid which slants on both sides, viz., towards the water and away from it. (1) Suppose the embankment to yield to the pressure of the fluid by turning round the otiinr edge A. Let ABCD be the cross-section of the embankment. Since the pressure of a fluid is always normal to the sur- face with which it is in contact (Art. 4), tlie pressure on the slanting face BC, of this embank- ment is inclined to the horizon, and hence the stability of tlie embankment is caused by its weight and the vertical pressure of the fluid on the face BC, while the effort to overthrow it is caused by the horizontal pressure of the fluid. Let P, and Pg be the horizontal and vertical components F E H B Fig. 12 I 30 EMBANKMENT WHEN ONE FACE IS SLANTING, of the normal pressure P, and « the angle which the direc- tion of the normal pressure makes with the horizon ; then we have, for the horizontal component, Pi = P cos a = area of BC x |CE x w^ cos « (Art. 15) = area of CE X i/iWi , where h = CE, and w^ is the weight of a cubic foot of the fluid. Similarly, Pg = area of BE x Ihw^ ; but area of CE is the projection of BC on CE, and area of BE is the projection of CB on EB ; i. e., the pressure exerted by a fluid in nny direction upon a surface is equal to the weight of a column of the fluid, whose base is the projection of the surface at right angles to the given direction, and whose height is the depth of the centre of gravity of the surface below the surface of the fluid. Hence, since the projection at right angles to the vertical direction is the horizontal projection, and that at right angles to a horizontal direction is a vertical one, we find the vertical pressure of the fluid against a surface by treat- ing its horizontal projection as Uie surface pressed upon, and, on the contrary, the horizontal pressure of the fluid in any direction by treating the vertical projection of the surface at right angles to the given direction as the surface pressed upon, and in both cases we must regard the depth of the centre of gravity of the surface below the surface of the fluid as the " height of the column." Let g, G, and g^ be the centres of gravity of AFD, FECD, and EBC ; let AB = «, DC = ft, AF = c, EB = d, and tv = the weight of each cubic foot of the embankment. The horizontal pressure of the water acting v.'- M tends to EMBANKMENT WHEN ONE FACE IS SLANTING. 31 turn the embankment over its outer edge A. Hence, we have the moment of P^ = Jih^Wi x ^h = ^¥w^ ; (1) the moment of Pg = dx^hiv^ x AH = \hdiv^ {a — ^d) ; the moment of APD = wt. of ADF x §AF = j^htv X |c = \c^hw ; the moment of FECD = wt. of FECD x (AF+ |FE) = Mwx(c+|J) ; the moment of EBC = wt. of EBC x (AB — |BE) = j^dhvj x(a — |rf) = Idhw (3a — 2d), .*. the moment of ABCD Ic+b) + ^ (3a - 2d)\ hw + ^ (3a - d) hw^. (2) If the embankment be upon the point of overturning on A, the moments in (1) and (2) are equal to each other, and we have ^h^w^ = [^ + ^(2c + 5)+^(3«-2rf)jAw+^(3a-«f)/mi VJ or, /*«= [2c2 + 3*(2c+*) + <^(3a-2rf)] — 4-e/(3a- or <\/[2c^-\-Sb {2c+b)-rd{^a-2d)]---^d {da— d). Cor. 1. — If the embankment is of the form of Fig. 11, ^ = 0, and (3) becomes 32 EMBANKMENT WHEN ONE FACE IS SLANTING, ' ^2= [2c2 + 3*(2c + J)];^, (4) which agrees with (5) of Art. 18. Cor. 2.— If the embankment is rectangular, c = 0, and (4) becomes w which agrees with (6) of Art. 18. (2) Suppose the embankment to yield to the pressure of the fluid by sUditiy along the horizontal base AB. The horizontal pressure of the fluid, from (1), is the vertical pressure of the fluid is The weight of the embankment is a-\-b 2 hiV'j and the entire vertical pressure of the embankment and the water on its face is a + * 2 hw -\- ^dhWf r= {aw ■\- hw ■\- diVi) IJi. Let fi = the coefficient of friction ; then the friction between the embankment and the surface of the ground on which it rests is (Anal. Mechs., Art. 92), {mo + biv + dw^) yifi. When the horizontal pressure of the water pushes the embankment forward, we must have \¥w^ = (aw -{- bw -^ dtOi) ^hfi ; PRESSURE UPON BOTH SIDES OF A SURFACE. 33 (4) and B of the ion on the or, more simply, 7i = (« + ft) \- d Lw , (6) and the dam will move or not according as 7t > or < {a + h)— ■\- dU. Cor. — If the embankment is rectangular, d z=0 and J = a, and (5) becomes h = 2a -- /I. 20. Pressure upon Both Sides of a Surface.— If a plane surface is subjected on both sides to the pressure of a fluid, the two resultants of the pressures on the two sides have £1 new resultant, which, as they act in opposite direc- tions, is obtained by subtracting one from the other. Let AB be a flood-gate with the water pressing on both sides of it, to determine the resultant pressure, and the centre of pressure. Let AB = «, the depth of the water on one side ; DB = h, the depth of the water on the otlier side ; P = the resulting pressure on the gate ; and w^ = the weight of a cubic foot of water. Then P = pressure on AB — pressure on DB ; .-. P = i (a« - ft2) wj. (1) Now let C and Cj be the centres of pressure of the sur- faces AB and DB, and Cg the point to which the resultant pressure P, is applied. Then, taking moments with respect to A, and putting ACg = "z, we have P xi = pressure on AB x AC — pressure on DB x AC, = ^%j X fa — iif^Wi (a - ^b). u EXAMPLES. !l 2a^ + 2ab — h^ 3 (a + J) (2) EXAMPLES. Fig. 14 1. The total breadth of a flood-gate is lb feet, and the depth is a feet ; the hinges are placed at d feet from the respective extremities of the gate; required the pressure upon the lower hinge. Let AB represent the height of tlie gate, D and E the hinges, and C the centre of pressure of the water. The pressure of the water upon each half of the gate = \a^bw ; and since the pressure of the Avater at C is supported by the hinges D and E, we have, by the equality of moments with respect to D, Pressure on E x DE = Pressure on C x DC ; but DE = a — 2rf, and DC = fa — a (Art. 21). If then be the vertex of the parabola, we shall have AD^ = ?fAO; W* AO = f AD^ which gives AO, and thus determines the quantity which flows out. If, however, AO be greater than AB, i. e., if be hclotv B, at 0', for instance, the surface of the liquid will 1 in BC, at P. We shaL then have. AD^ = ^AO to (2) (3) rti- to I [lich clow e in STREI^OTff OP PIPES AND BOILERS. and BP* = ^BO'; which determines the position of P. (Besant's Hydrostatics, p. 154.) 2. A straight tube AB, filled with liquid, is made to rotate about a vertical axis through A ; find how much flows out at B. Ans. AH above P, where P is tangent to the parabola whose latus rectum is -^ and whose axis is coincident with the vertical line through A, and AP = — „ cot « cosec a, w« where « is the angle OAB. Fig. 18 28. Strength of Pipes and» Boilers.— An important application of the theory of the pressure of fluids is the determinai)ion of the thickness of />*j9es, boilers, etc. In order that these vessels shall be trong enough to resist the pressure of the liquid, their walls must be made of a certain thickness, which depend? upon the pressure of the liquid and the internal diameter of the vessel. Let it he required to find the thickness of a pipe of any Vfiaterial necessary to resist a given pressure. A cylindrical vessel may burst either transversely or lon- gitudinally ; but the former is less likely to occur than the latter, as appears from the following investigation. (1) When the rupture is transverse. Let ABCD (Fig. 19) be a section of pipe perpeindicular to its axis, the interior surface of which is subjected to a f 40 STRENGTH OF PIPES AND BOILERS. pressure of jo on each unit of surface. Let 2r be the diam- eter MD of the interior, then will the surface i)ressed be measured by Trr2, which is the area of the cross-section of the interior, and the whole pressure upon the surface of the end of the pipe and which produces rup- ture will be measured by Trr»j». (1) Let e = AE = the thickness of the pipe ; then the cross-section of the mate- rial of the pipe = TT (r + e)2 — Trr* = -ne {e + 2r). Let T denote the strength of the material of which the pipe is composed, for each unit of cross-section ; then the strength of the entire pipe in the direction of the axis = ne{e + 2r) Ty (2) and since the whole pressure in (1) when rupture is about to take place must be held in equilibrium by the strength in (2), we have ire {e -{- 2r) T = 'nr^. e = rp 2f' since e is usually very small in comparison with 2r. (3) (2) Wlien the rupture is longitudinal. Let EMH be any portion of the wall whose length is I, and let 2a = the angle ECH. Then, since the projection of EMH at right angles to the line MD passing through the centre is a rectangle whose area = 2rZ sin «, the mean pressure of the fluid on the wall, EMH = 2W sin «p (Art 19). (4) STRENOTH OF PIPES AND BOILERS. 41 be the the (2) out gth (3) is I, bion ugh lean (*) Now this pressure must be held in equilibrium by the forces of cohesion, R, R, acting tangentially on the cross- sections, AE and BH, of the wall of the pipe. Denoting the components of iZ, /^, parallel to MD, by Q^ Q, we have 2Q = 2R sin a = 2elTsm a, (5) e being the thickness of the pipe and T the strength of each unit of section. Therefore, from (4) and (5) we have, 'itelT am a = 2rlp sin « ; 0-^ 0- rp> (6) which shows that the thickness of the pipe is independent of its length. Otherwise thus, by the principle of work. The whole surface of the interior of the pipe = "Z-rrrl ; and the whole pressure upon the surface = )l-nrlp. Suppose the pipe to rupture longitudinally,* under this pressure, its radius becoming r-f e?r; then the path described by the pressure will be dr, and the work done by the pressure = ^nrlpdr. (7) The force R, which resists rupture and acts tangentially, = eTl While the radius of the interior changes from r to r + dr, the circumference changes from 2nr to 2n{r-\-dr); then the path described by the resistance = 2n dr, and the work done by the resistance = 2-neTldr. (8) * Longitudinal tension prmluces transverse rupture, and transverse tension pro- duces longitudinal rupture. The stretching tendency to rupture longitudinally is a transverse stretching, i. «., the pipe tends to bulge out all along its length ; hence, traosveraely, r becomes r-¥dr. 49 EXAMPLES. Therefore, from (7) and (8), by the principle gt work, we have 2neTl dr = %-nrlp ^r, which is the same as (6). From (3) and (6) it follows that, to prevent a longi- tudinal rupture, the ivall must be made twice as thick as would be necessary to prevent a transverse one. Cor.— Since p .- ziv [from (1) of Art 10], (3) and (6) become, respectively, rp rzw and rp _ rzw ^ ^ — W — ~m~ 'i that is, the thickness of similar pipes must vary di- rectly as their diameter and us the pressure upon the unit of surface, or in the case of a liquid, as the depth of the pipe below the upper surface of the liquid, and inversely as the strength of each unit of section. A pipe which has twice the diameter, and has to sustain four times the pressure of another, must be eight times as thick. (See Weisbach's Mechs., Vol. I., p. 739 ; Bartlett's Mechs., p. 294; Tate's Mechs., p. 268.) EXAMPLES. 1. It is found that the pressure is uniform over a square yard of a plane area in contact with Buid, and that the pressure on the area is 13608 lbs.; find the measure of the pressure at any point (Art. 6), (1) when the unit of length is du iucb, (2) when it is two inches. Ans. (1) 10^ lbs. ; (2) 43 lbs* EXAKPLES. e 2. If the area of a (Fig. 4) be a square inch, and if it be pressed by a force of 15 lbs., what pressure * will this trans- mit to the piston A if its diameter be 10 in. ? Arts. Pressure on A = 1178 lbs. 3. If the diameter of a be 4 in., and if the pressure on it be 185 lbs., what pressure will be exerted on A if its area is one stjuare foot? Ans. Pressure on A = 2120 lbs. 4. It* ilie area of a be 20 square inches, and if it be pressed by a force of 360 lbs., find the diameter of A so that it shall be pressed upwards by a force of 10 tons (one ton = 2240 lbs.) Ans. Diameter of A = 39.8 in. 5. If the diameter of A (Fig. 3) be one inch, and if the surface at E be a square whose side is one-quarter of an inch, find the pressure transmitted to E if that on A be 10 lbs. Ans. Pressure on E = 0. 795 lbs. 6. If the area of A be 2^ sq. in., and the pressure on it 56 lbs., find the pressure transmitted to a surface at B, the area of which is a triangle whose base is | of an inch, and whose height is -^ of an inch. Ans. Prrssure on E = 0.42 lbs. 7. A cylindrical pipe which is filled with water opens into another pipe the diameter of which is three times its own diameter ; if a force of 20 lbs. be applied to the water in the smaller pipe, find the force on the open end of the larger pipe which is necessary to keep the water at rest. Ans. 180 lbs. 8. Kequired (1) the pressure on the sides of a cubical vessel filled with water, and (2) the pressure on the bottom, the side of the vessel being a ft. (Art. 10). Ans. (1) 125aMbs.; (2) 62.5«Mbs. 9. A cylindrical vessel is filled with water ; tlie height of the yessel is a ft, and the diameter of the base d feet. Find (1) the pressure upon the side and (2) the pressure on the bottom. Ans. {\) ^\\-rT(M ) {ii) Ih^nacK * lo the first seven examples, t}io weight of tbc liquid Itself la not ooneidered. 44 EXAMPLES. 10. Find the height of the vessel in Ex. 9 so that the pressure on the side may be equal to the pressure on the bottom. Ans. The height must equal the radius of the base. 11. The pressure on a square inch of surface in a vessel of mercury is 1000 grains. Find the pressure on a circular surface of one-quarter inch radius, placed 9 in. lower down, mercury being 13.5 times as heavy as water. Ans. Pressure = 0.8886 lbs. 13. The water in a canal lock rises to a height of 18 ft. against a gate whose breadth is 11 ft. Find the total press- ure against the gate. Ans. Pressure = 49|^ tons.* 13. The upper side of a sluice-gate is 10| ft. beneath the surface ; its dimensions are 3 ft. vertical ^>y 18 in. horizon- tal. Find the pressure upon it. Ans. Pressure = 1\ tons.* 14. A dyke to shut out the sea is 200 yards long, and is built in courses of masonry one foot high ; the water rises against it to a height of 6 fathoms. Find the pressure against the 1st, 18th, and 36th courses. i 1st pressure = 610.4 tons.* Ans. I 2d pressure = 318.1 tons. ( 3d pressure = 8.6 tons. 15. Find the pressure, in pounds, of a cylinder of water 4 inches in diameter and 45 ft. in height. Ans. Pressure = 244.8 lbs. 16. A cubical vessel, each side of which is 10 ft., is filled with water, and a tube 32 ft. long is fitted to an aperture in it, whose area is one square inch. If the tube be vertical, and of the same size as the aperture, and filled with water, find the pressure on the interior surface of the vessel, (1) neglecting the weight of the water it contains, (2) when the weight of the water is taken into account. Ans. (1) 1,200,000 lbs. ; (2) 1,387,500 lbs. *ODeton = 2M01b«. EXAMPLES, 45 he ;be ' i 17. Find the pressure ou a square inch at a depth of 100 ft. in a lake, (1) neglecting, (2) taking account of the r.i mospheric * pressure. Ans. (1) 43|f ibs. ; (2) 58 lbs. 18. A reservoir of water is 200 ft. above the level of the ground floor of a house ; find the pressure of the water, per square inch, in a pipe at a height of 30 ft. above the ground floor, neglecting atmospheric pressure. Ans. 73|Jf lbs. 19. An equilateral triangular area is immersed vertically in water with a side, one foot in length, in the surface. Find the pressure upon it in ounces. Ans. 125 oz. 20. A hollow cone, vertex upwards, is just filled with liquid. Find (1) the pressure on its base, (2) the normal pressure on its curved surface, (3) the vertical pressure on the curved surface. [Let r = the radius of the base and h = the altitude.] Ans. {l)gp7Ti^h; (2) IgpnrhVi^^; (3) li/pr.r'A. 21. A vertical rectangle has one side in the surface of a liquid. Divide it by a horizontal line into two parts on which the pressures are nqual. Ans. If h be the vertical side, the depth of the horizontal A. ^2. A vertical triangle, altitude h, has its base horizontal ?.nd its vertex in the surface. Divide it by a horizontal line into two parts on which the pressures are equal. Ans. The depth = -— • V 2 23. A smooth vertical cylinder one foot in height and one foot in diameter if. filled with water, and closed by a he^vj piston weighing 4 lbs. Find the whole pressure on its curved surface. . ^_ 1257r ., Ans. IG H -7— lbs. line = • See Art. 11, Cor. 9- 46 EXAMPLES, 24. A hollow cylinder, closed at both ends, is just filled with water and held with its axis horizontal ; if the whole pressure on its surface, including the plane ends, be three times the weight of the fluid, compare the height and diameter of the cylinder. Ans. As 1:1. 25. The side AB of a triangle ABC is in the surface of a fluid, and a point D is taken in AC, such th»t .the pressures on the triangles BAD, BDC, are equal. Find the ratio AD : DC. Ans. As 1 : V2 — J. 26. The diameters of the two pistons, p and P (Fig. 4), are 2i^ in. and 9 in., respectively, and the smaller is 60 in. above the larger. What force must be applied to the smaller piston that the larger may exert a pressure of 1600 lbs.? Am. 112.8 lbs. 27. Compare the pressure on the area of a parabola with that on its circumscribing rectangle, both being immersed perpendicularly to the vertex. Ans. As 4 : 5. 28. A cubical vessel is filled with two liquids, of given densities, the volume of each being the same. Find the pressure on the base and on any side of the vessel. Let flf be a side of the vessel, p and p' the densities of the upper and lower liquids, p' being greater than p. The pressure on the base = the weight of the whole fluid The pressure on the upper half of any side To find the pressure on the lower half, replace the upper liquid by an equal weight of the lower liquid, which will not affect the pressure at any point of the lower half. If a' be the height of this equal weight, we have EXAMPLES, ♦7 r I f^ pa =P2' and the depth of the centre of gravity of the lower half below the upper surface of the equal weight = "' + 1 = 1(^ + 1)' therefore, the pressure on the lower half 2p^ ,fl' a/ ^ + 1) J p /^" LI — " ■*- nV •< VhX D\0G 1 3 Fig. 20 = ^«^(p' + 2p). (Besaiit's Hydrostatics, p. 37.) 29. A circle is just immersed vertically in a fluid. Find on which chord, drawn from the lowest point, the pressure is the greatest. [Let ADBC be the circle with radius a and BC the required chord, which bisect in H, and draw HK perpendicular to AB; ,*. etc.] Ans. AK = fa. 30. A semicircle is immersed vertically in a fluid, with its diameter in the upper surface; find on which chord, parallel to the surface, the pressure is the greatest, supposing the density of the fluid to increase as the depth. [Let LBM (Fig. 20) be the semicircle, and DE the chord on which the pressure is the greatest, and a the radius of the circle. Then if the density were uniform, the pressure would vary as DG x GF (Art. 15) ; but, since the density varies as the dex)th, the pressure varies as DG x Gi"* ; ••• etc.] A71S. FG = aVi 31. If LBM (Fig. 20) be a parabola, FB = ft. the latns rectum = 4rt, and the other conditions the same as in Ex. 30, find FG, the depth of the chord of greatest pressure below the upper surface. Ans, FG = |ft. 48 EXAMPLES. 32. The lighter of two fluids, whose densities are as 2 : 3, rests on the heavier, to a depth of 4 in. A square is im- mersed in a vertical position, with one side in the upper surface. Determine the side of the square in order that the pressures on the portions in the two fluids may be equal. Ar.cs. f (l + Vio) in. 33. Find the centre of pressure of a semi-parabola, the extreme ordinate coinciding with the surface of the fluid. [Let LBF (Fig. 20) be the semi-parabola; let BF = «, and LF = h, and suppose to be the centre of pressure, OG being parallel to LF.] Ans. FG = 4a ; GO = ^b. 34. A o'ladrant of a circle is just immersed vertically in a liquid, with one edge in the surface, as in Ex. 3, Art. 16. Find the centre of pressure when the density varies as the depth. Taking the edge in the surface for the axis of y and the vertical edge for the axis of x, we find X = 32a. 15 tt' - 16 a 35. The total breadth of a water passSige closed by a pair of flood-gates is 10 ft. and its depth is 6 ft. ; the hinges are placed at one foot from the top and bottom. Find the pressure upon the lower hinge when the water rises to the top of the gates. Ans, 4218f lbs. 36. If we suppose everything to be the same as in Ex. 2, Art. 20, except that the height of the wall is determined by the condition that the wall just sustain the pressure when the water rises to the top, what is the height of the wall ? Ans. 6.96 ft. 37. A wall of masonry, a section of which is a rectangle, is 10 ft. high, 3 ft. thick, and each cubic foot weighs 100 lbs. Find the greatest height of water it will sustain with- out being overturned. Ans. 6'v^2. I EXAMPLES, 49 38. If the height of thft wall be 8 ft, its thicknjss 6 ft-., and each cubic foot weighs 180 lbs., find whether it will stand or fall when the water is on a level with thd top. Ana. 39. The depth AB of the water in the head bay (Fig. 13) is 7 ft., the depth DB of the water in the chamber of the lock is 4 ft., and the width of the lock-chamber is 7.f» ft. ; find (1) the resultant pressure upon the gate AB, and ('2) the depth of the point of application of the resultant, press- ure below the surface of the water in the head bay. Ans. (I) 7734.4 lbs.; (2) 4.18 ft. 40. If the vessel (Fig. 15) make 140 turns per minut<^, find the value of NM. Ans. 1.78 in. 41. A hollow paraboloid of revolution, with its axis ver- tical and vertex downwards, is half filled with linuid. With what angular velocity must it be made to rotate about its axis, in order that the liquid may just rise to the rim of the vessel. Ans. If 2j» = latus i-ectum. w* = 42. If the vessel in th& last example be tilled with liquid, find the angular velocity and the time of rotation that it may just be emptied. -9. =-\/f Arts. If 2» = latus rectum, w* = ^: time = 43. A hemispherical bowl is filled with liquid, which is made to rotate uniformly about the vertical radius of the bowl. Find how much runs over. . 1 ixa^ul^ Ans. -: • * g 44. A closed cylindrical vessel, height h and radius a, is just filled with liquid, and rotates uniformly about its ver- tical axis. Find the pressures on its upper and lc«ver ends, and the whole pressure on its curved surface. \i CHAPTER II. EQUILIBRIUM OF FLOATING BODIES. GRAVITY — SPECIFIC of ij 24. Upward Pressure, Buoyant Effort.— To find the resultant pressure of a liquid on the surface of a solid either wholly or partially immersed. Let ABCD be a solid floating in a liquid whose upper surfflot is EF. Imagine this solid removed, and the space it occupied filled with the liquid, and suppose this liquid to be solidified. It is clear thiiv the result- ant pressure upon this solidified liquid will be the same as upon the original solid. But this solidified mass is at rest under the action of its own weight and tiie pressure of the surrounding liquid ; and, as its own weight acts vCi'tically downward through its centre of gravity, the resultant pressure of the surrounding liquid must be equal to the weigut of the solidified mass, and must act ver- tically upwards in a line passing tbiough its centre of gravity. The above reasoning is equally applicable to the case of a bodv immersed in elastic fluid. Therefore, if a solid be either wholly or partially im- mpTsed in a fluid, it loses as much of its weight as is e(/u(d to the weight of the fluid it displaces.* * The diflcovery of this jirincipltf is due to Archimedes. (Goodeve, p. 190 ; Gal- braith, p. 48.) Fifl.2J UPWARD PHESaUREi UUOYAaM' EFFORT. 51 Cor. 1. — If a body be supported entirely by a fluid, the weight of the body must be equal to the weight of the fluid displaced, and the centres of gravity of the body and of the fluid displaced must lie in the same vertical line. =', ScH. — These conditions hold good, whatever be the nature of the fluid in which the body is floating. If it be hetero- geneous, the displaced fluid must consist of horizon tttl strata of the same kind as, and continuous with, the horizontal strata of uniform density', in which tlie particles of the sur- rounding fluid are necefa^^rily arranged. If, for instance, a Bolid body float in water, partially immersed, its weight will be equal to the weight of the water displaced, together with the weight of the air displaced. The upward pressure of a fluid against a solid, and which is equal to the weight of tlie displaced fluid, is called the buoyant effort of a fluid. The centre of gravity of the dis- placed fluid is called the centre of buoyancy. The buoyant effort exerted by a fluid acts vertically upwards through the centre of buoyancy. The enunciation and proof of this proposition are due to Archi- medes, and it is a remarkable fact in the history of science, that no further progress was made in Hydrostatics for 1800 years, and until the time of Stevinus, Galileo, and Torricelli, the clear idea of fluid action thus expounded by Archimedes remained barren of results. An anecdote is told of Archimedes, which practically illustrates the accuracy of his conceptions. Hiero, king of Syracuse, had a certain quantity of gold made Tuio a ciy)wn, and suspecting that the goldsmith had abstracted bume of the gold and used a portion of alloy of the same weight in its place, he applied to Archimedes to investigate whether such was the case, and to ascertain the nature of the alioy. It is re- lated that while Archimedes was in his bath, reflecting over the diffi- cult problem which the king had given him, he observed the water running over the sides of the bath, and it occurred to him that he was disi>lacing a quantity of water equal in volume to that of his own body, and therefore that a quantity of pure gold equal in weight to the crf^wn would displace less water than the crown, the volume of imy weight of alloy being greater than that of an equal weiglrt of gdd. bi EQUILIBRIUM OF AN LMMERSED SOLID. He concluded at once that he could completely solve the king's prob- lem, by weighing the crown in water. Overjoyed with his discovery, he ran directly into the street, crying out, " Eureka I Eureka ! " The two books of Archimedes which have come down to us were first found in old Latin MS. by Nicholas Tartaglia, and edited by him in 1537. These books cont&in the solutions of a number of problems on the equilibrium of paraboloids, and various problems relating to the ecjuilibrium of portions of spherical bodies. The authenticity of these books is confirmed by the fact that they are referred to by Strabo, who not only mentions their title, but also quotes from the first book. 25. Conditions of Etiuilibrium of an Immersed Solid. — Let v denote the volume and p the density of the solid ; v' the volume and p' the density of the displaced fluid : the weights of the solid and of the displaced fluid will be respectively gpv and gp'v' ; then, if the solid rest in equilib- rium in the fluid, we shall have gpv = gp'v'. (1) If we suppose the solid to be entirely immersed, the vol- umes V and v' will be equal, and the densities p and p' must also be equal if the solid remains in equilibrium, having no tendency either to ascend or descend. But if the weight of the immersed solid be greater than that of the fluid displaced, we shall have gpv > gp'v and the solid will be urged downwards by a force equal to gpv — gp'v. If, on the contrary, the weight of the solid be less than that of the fluid, we shall have gpv rob- rory, they also (1) That is, the wholly innnersed solid will descend, re- main at rest, or ascend, according as its density is greater than, equal to, or less than the density of the fluid. In the first case the solid will descend to the bottom, and press it with a force equal to the excess of its weight above that of an equal bulk of fluid. In the third case the solid will rise to the surface, and be but partially immersed, the volume v' of the fluid displaced by the solid having the same weight as the entire solid. [An Qgg, placed in a vessel of fresh water, sinks to the bottom of the vessel, its mean density being a little greater than that of the water. If, instead of fresh water, salt water is employed, the egg floats at the surface of the liquid, which is a little denser than the egg. If fresh watrr is carefully poured on the salt water, a mixture of the two liquids takes place where they are in contact ; and if the egg is put in the upper part, it will descend, and, after a few oscillations, remain at rest in a layer of liquid of which it displaces a vehime whose weight is equal to its own.] Cor. — Prom (1) we have V : v' '.: p' : p\ therefore, // a homogeneous solid float in a fluid, its whole volume is to the volume of the displaced fluid as the density of the fluid is to the density of the solid. ScH. — When the floating solid and fluid are both homo- geneous, the centre ot gravity of the part immersed will coincide with the centre of buoyancy. The section of a floating body formed by the plane of the surface of the fluid in which the body floats is called the plane of flotation. The line passing through the centre of 64 EXAMPLES, gravity of the floating body and tho centre of buoyancy is called the axis of flotation. The weight gpv of the body acting downwards, and the buoyant effort yp'v' acting upwards (Art. 24, Sch.), form a couple, by which the body rotates till the directions of these forces coincide, i.e., till the centre of gravity of the body and the centre of buoyancy come into the same vertical line. EXAMPLE. 1. A piece of oak containing 32 cubic inches, floats in water ; how much water will it displace, the density of the oak being 0.743 times that of water ? Ans. 23.776 cu. in. tJG. Depth of Flotation.*— T/ X = a^^-r 3. Let the body be a sphere of radius a, floating in a fluid. Kecjuired the depth of flotation. Here the displaced fluid has the form of a segment of a sphere ; hence, calling x the depth, we have, from mensura- tion, v' = TTx^ {a — \x)f and V = f^raSj - — 3a;=^ {a — \x) V ~ 4a' = ^, [fTom(l)]; P we have, therefore, to solve a cubic equation in order to find the depth of flotation of the spherct 66 EXAMPLES. 4. Let the body be a cylindrical pontoon,* with plane ends, and having its axis horizontal. Required to find the load requisite to sink the pontoon to a given depth. Let AD be the intersection of the plane of flotation with the end which is a right section. Put A = the area ADK, the plane surface of immersion, and / = AB, the length of the cylinder ; and let W = the required load that will sink it to the depth HK. Then, calling p' the density of the fluid, we have -«^ and volume of displaced fluid = Alf weight of displaced fluid = gp'Al; .'. W = gp'Al. (1) (2) A may be found as follows: let r = CK, and = angle ACK ; then we have., from mensuration. 180 + I ''"4 A = r^ln- which in (2) gives, which is the required load. Cor. 1.— If e = 165°, we have, from (4), W = igpr^ (V^ + 1) i- (3) W (5) * Pontoons are imrtable boatH, covered with balkt*, planks, etc., for formiuf; floating b. idges over rivcro. They are now usually made of tin, in the ^bnpe of a cylinder, with hemispherical ends. (Tate'e Meclianical Philosophy.) |ne he 30, BTABILtTY OP BqvrUBRWM. 5^ Cor. 8. — If the fluid be water, (5) becomes fT = ir2 (Y^ + 1) 1 63.5 (Art. 10, Cor. 1). (6) 5. Let the body be a cone floating with its base under the fluid, and the axis a vertical. Find the depth of flota- tion. 3/ -jj Ans. a — a\/ 1 , • 6. A man whose weight is 150 lbs. and density 1.1, just floats in water by the help of a quantity of cork. Find the volume of the cork in cubic feet, its density being .!&4, call- ing the density of water 1. Ans. t^V? of a cubic foot. 27. Stability of Equilibrium — If a floating body is in equilibrium, the centres of gravity and of buoyancy are in the same vertical line (Art. 24, Cor. 1). Imagine the body to be slightly displaced from its position of equilibrium by turning it round through a small angle, so that the axis of flotation shall be inclined to the vertical. If the body on being released return to its original position, its equilibrium is stable ; if, on the other hand, it fall away from that posi- tion, its original position is said to be one of unstable equi- librium ; when the body neither tends to return to its original position, nor to deviate farther from it, the equilib- rium is said to be one of indifference. The investigation of this problem in its utmost extent would lead to very tedious and complex operations, which would clearly be beyond the limits of this treatise ; we shall therefore premise the thre following hypotheses, in order that we may obtain comparatively simple results: 1. The floating body will be regarded as symmetrical with respcit to a vertical plane through its centre of gidvity when the whole is at rest, so that we need consider only the problem for the area of a phne section of the body. 58 STABILITY OF EQUILIBRIUM. Fig. 23 2. The displacement will be regarded as very small. 3. The vertical motion of tiio centre of gravity of the body will be disregarded, as indefinitely small. Ijct EDF represent a body which has changed from its upright to its present inclined position, by turning tiirough a small angle ; let ABD repre- sent the immersed part of the body Iwjfore displacement, and HKD that immersed after dis- placement, and G and the centres of gravity and of buoyancy before displacement. While the body moves from its upright to its inclined position, its centre of buoyancy moves from to 0', which latter is in the half of the body most immersed, and the wedge-shaped part ACH passes up out of the water, drawing the wedge-shaped part BCK down into it. Let the vertical lino through 0' meet GO in M. Now since the buoyant effort is equal to the weight of the whole solid (Art. 24, Sch.), the magnitude of the part im- mersed will be unaltered ; therefore ABD = HKD, and ACH = BCK ; also, the buoyant effort P, acting at 0' vertically upwards, and the weight P of the solid, acting at . G vertically downwards, form a couple which tends to restore the body to its original position when M is above G ; and, on the contrary, it tendt' to incline the body farther from its original position when M is helow G. Hence, the stability of a floating body, a ship, for instance, depends upon the position of the point M, where the vertical lino through the centre of buoyancy, in the inclined position of the body, cuts the line connecting the centre of gravity and centre of buoyancy in the upright position of the body. The position of the point M will in general depend on the extent of displacement. If the displacement be very small, STABTLITT OF SQVILIBRTUM. 59 t. e., if the angle between GO and the vertical be very small, the point M is called the metacentrc, and the question of stability is now reduced to the determination of this point. A ship, or any other body, floats with stability when its metacentre lies above its centre of gravity, and without sta- bility when it lies below it; it is in indifferent equiUbrium when these two points coincide. Hence the danger of taking the whole cargo out of a ship without putting in ballast at the same time, or of putting the heaviest part of a ship's cargo in the top of the vessel and the lightest in the bottom, or the risk of upsetting .when seveml people stand up at once in a small boat. One of the most important problems in naval architecture is to secure the ascendancy, under all circumstances, of the metacer.tre above the centre of gravity. This is done by a propel form of the midship sections, so as to raise the meta- centre as much as possible, and by ballasting, so as to lower the centre of gravity.* The horizontal distance MN, of the metacentre M, from the centre of gravity G of .he body, is the arm of the couple whose forties are P and P, the weight of the body and the buoyant effort; and the moment of this couple, which measures the stability of the body, is P* MN. Let GM = c, and the angle OMO', through vhich the body rolls, = 0, and denote the measure of the stability by S'y then we have S = p. MN = Pc sin e ; (1) therefore, the stahility of a body, in general, varies as its weight, as the distance of its metacentre from its centre of gravity, and, as the angle of inclination ; and, hence, in the same body, for a, given inclination, it depends only upon the distance of its metacentre fromt its centre of gravity. * Besaot^H HydroBtatics, p. 66. 60 POSITION OF THE MJSTACENTRE. 28. The Position of the Metacentre ; the Measure of the Stability.— Since the stability of a body depends principally upon the distance of the metacentre from the cenlre of gravity of the body, it becomes iniportt?nt to de- termine the position of the metacentre. Let A = the cross-section ABD = HKD (Fig. 23) of the immersed part of the body (Art. 27), and A^ = the cross-section ACH := BCK ; let (j and g' be the centres of gravity of ACH and BCK ; let rt = the horizontal distance RL, between these centres of gravity, and s = the horizon- tal distance between and 0', the centres of buoyancy. Then, taking moments round G, we have, HKD X MN - ACH X RN = ABDxNT + BCKxNL; or, and A (MN - NT) = J, (RN -\- NL); OM = -^' = -Ai^ sin A sin 0* which is the height of the metacentre above the centre of buoyancy. Let 60 = e; then c = GM = e -f- A^a A sin e* 0) irhich gives the height of the metacentre above the cen- tre of gravity. Substituting this value of c in (1) of Art. 27, we get S= P (^- 4- e sin o), (2) which is the measure of the stability. i Measure ot stability. 61 If the point were below G, e would be negative and (3) would be Hence, in general, we have sin A (3) 8=p{^±e%m (4) the upper or lower sign being used according as the centre of buoyancy is above or below the centre of gravity. Coic. 1. — If the displacement be small, the cross-sections ACH and BCK can be treated as isosceles triangles, and Fin = $. Denoting the width AB = HK of the body at the plane of flotation by b, we have and which in (4) gives RL = « = |>, (5) OoR. 2. — When the centre of buoyancy is above the cen- tre of gravity of the body, the stability is positive, as also in the case when the centre of buoyancy is below the centre of gravity whi^e e ia less than r^^ ; in this case the equi- librium is that of stability. If e is greater than ^^^ , and the centre of buoyancy is below the centre of gravity of the body, the stability is neg- ative, or the equilibrium is that of instability. If e is negative and equal to . j-. , the stiibility is zero, and the equilibrium is that of ndifference. That is, the centre of buoyancy may be below the centre of gravity and yet the stability be positive, so long as e does 62 EXAMPLES. not exceed irg-j, which term is always *^e distance be- iTcen the metaccntre and the centre of bu yancy. If the centre of gravity of the body coincides with the centre of buoyancy, we have e = 0, and (5) becomes S= P 12.4 e. (6) Hence, generally, the stahility is positive, negative, or zero, according as the metacentre is above, beloir, or coincident with the centre of gravity of the floating body. A vertical line O'M through the centre of buoyancy is called a line of support. Cor. 3. — From the above results we see that the stability of u body is greater the broader it is and the lower its centre of gravity is. (See Weisbach's Mechs., Vol. L, p. 760 ; also Bland's Hydrostatics, p. 120.) EXAMPLES. 1. Determine the stability of a homogeneous rectangular parallelopipod floating in a fluid. Let HR be the line of flotation of a vertical section passing through the centre of gravity G ; let h ■= the breadth EF of the section of the par- allelopipod, h = the height EC, and y =; the depth of immersion AC. Then we have A = by, and e = — i (A — y), H'K -L A r>^ t "TPtf^ ^C7 J /te§ '>ji = -=L -Z^ -H 3^r--^ = — — i:_ — -— - ;^=ir- — — — ■- — Fig. 24 e being negative since the centre ov buoyancy is below th centre of gravity. Substituting in (5), we have EXAMPLES. 68 ^=^(ife-M)»- (1) Let the density of the material of the parallelopiped be p times that of the fluid ; then (Art. 25, Cor.), p \ \ \: y : h'y .-. y = hpy which in (1) gives *=[i2^-t(l-'')]^''' ivhich is the measure of the stahiUty required. (8) Cob. 1. — To determine the limits of stability depending upon the dimensions and density of the solid, let >$ = 0, and (2) becomes *»-6A2p(l-p) = 0; (8) or. I = Viip (1 - P). If p = ^, we have b h = iV6 = 1.225, and hence in this case the parallelopiped floats in stable, indifferent, or unstable equilibrium, .iccording as the breadth is >, =, or < 1.225 times the height. Cor. 2. — Solving (3) for p, we get __ 1 ^ 1 /:, %W which is real when y^ is = or < \V^ \ *• «•» when the ratio of the breadth of the solid to the height is equal to, or 64 E2-AMPLES. less than \y/^h two values may be assigned to the density of the solid whic' will se 't to flo>U in indifferent equi- librium. If, for instance, L P = i ± iVl -- f A, ^\'" have 0.788C8 or 0.21132. Cor. 3. — "When j > |\/C, the value of p is imaginary, i. e., if the ratio of the breadth of the solid to the height is greater than i^V^G, nc> Nalue can be given to the density which will cause the stability to vanish. In this case tlio solid, placed with EF horizontal, must in all cases continue to float permanently in that position, whatever may be the density, providing it is always less than that of the fluid. Cor. 4. — The tei*n -.. '» (!)> or --75- in (2), is the dis- 12y ^ ' VZhp ^ ' tance between Hio oontve of buoyancy and the metacentre. 2. Determine the angle of inclination B, in order that the parallelopiped EFDC may be in a position of indiflfer- ent equilibrium. Let h = the breadth EP of the section of the paral- lelopiped, y = the depth of immei'sion AC = BD, and = angle AOH. Then A = ABDC = HKDC = h> (1) A, = AOH = BOK. and AH = BK = idtanO; But AO = OB = ^b, therefore ^j =1*2 tan e. J w EX.iMPLES. 65 Tjet g and g' bo the centres of gravity of the triangles AOII and BOK ; draw M^ parallel to AH, and glX and MQ perpendicular to HO. Then M V^p(l — p), the value of tan is imaginary, i. c, if the ratio of the breadth to the height is greater than ^%f,[\—p), no value can be found for the inclination which will cause the stability to vanish. (Compare with last example.) 3. T* the breadth of the parallelopiped is ecjual to its height, and if p = I, find the inclination 0, that tlie paral- lelepiped may float in indifferent equilibrium. Ans. e = 45° 29. Specific Gravity.— TAe apecific gravity of a body is the ratio of its ireight to the weight of an equal volume of some other body taken as the standard of comparison. The density of a body has been defined (Anal. Mechs., Art. 11), to be the ratio of the mass of the body to the mass of an equal volume of some other body taken as the stand- ard ; and since the weights of bodies are j>roportional to their masses, it follows that the ratio of the weights of two bodies is equal to the ratio of their masses. Hence, the measure of the specific gravity of a body is the same as that of its density, provided that both be referred to the same standard substance. Thus, let *S', W, V, and p be the specific gravity, weignt, volume, and density, respectively, of one body, and S\, PT,, F,, and pj the same of another body; then we have W _ gpV _ pV W~^-9P^V,-p,V,' i& a! ai (1) THE STANDARD TEMPERATURE, and making the volumes equal, we have 67 (2) that is, the ratio of the specijie gravities of two bodies is equal to that of their densities. Now suppose the body whose weight is W, to be assumed as the standard for specific gravity ; then will S^ be unity, and (2) will become A'=-?- = -^. (8) Also, if the same body be assumed as the standard of density, Pj will be unity, and (3) will become 6' = -^ = p. W Hence, the Tneasure of the specific gravity of a body is the same as that of its density, i. e., the nambers S and p are identical, when both specific gravity and density are referred to the same substance as a standard. 30. The Standard Temperature.— The standard sub- stance to which specific gravity and density are referred is not necessarily the same, and therefore 8 and p will in gen- eral be different numbers. In practice, it is usual to adopt water as the standard in determining the specific gravities of solids and incompressible fluids ; and for the purpose of rendering the comparison more exacf, the water must first be deprived by distillation of any impurities which it may contain. The dimensions of all bodies being more or less changed by changes of temperature, it becomes necessary to adopt a standard temperature at which experiments for determining 68 TffB STANDARD TEMPSRATURE, specific gravities must be performed. The English ♦ usually take fur this purpose the temperature of 00° Fahrenheit, it being easily obtained at all times, and the tables of specific gravities are usually given with reference to distilled water at this temperature as the standard. When the experiment cannot be performed at the standard temperature, the result obtained must be reduced to what it would be at this tem- l)orature, i. e., the apparent specific gravity, as obtained by means of water when not at the standard temperature, must be reduced to what it would have been if the water had been at the standard temperature. Thus, let p be the density of any solid, S^ its apparent specific gravity as obtained by water when not at the stand- ard temperature, and pj the corresponding density of the water ; and let S be the true specific gravity of the body as determined by water at a standard temperature, the corre- sponding density of the water being pj. Then, from (3) of Art. 29, we have 8,=. __ P_ Pi and s = p.' S '*». ^i" Pi (1) Calling the density of the standard temperature unity, (1) becomes S = ^,p,. (2) That is, the specific gravity of a body as dctenmncd at the standard temperature of the water is equal to its specific gravity determined at any other tempera- ture, midtiplied by the density of the water at this temperature, the density of the water at the standard temperature being regarded as unity. ScH. — In the cases that occur most frequently in prac- tice, such nicety is unnecessary, and the experiment may be * The French asnally take the tempentare at wbicb water has its maximam of deneitj, wbicb la 89<>.4 F. METHODS OF FISDINQ SPECIFIC OR A VITT. 69 performed with water at any temperature; but the temper- ature must be noted and a correction applied for it which depends upon the density of water at the experimental temperature.* The weight of a cubic foot of distilled water at the stand- ard temiKjrature is 1000 ozs. = 62| lbs. ; hence we find the weight of a cubic foot of any substance in ounces or pounds by multiplying its specific gravity by 1000 or 02^. It appears, therefore, that by means of the specific gravi- ties of homogeneous bodies, their weights may be deter- mined without actually weighing them, provided their volumes are known ; and conversely, however irregular the shape of bodies may be, if their weights and specific gravi- ties are known, their volumes may be determined, viz., by dividing the weight by the specific gravity. Tlie specific gravities of gases and vapors are usually determined by referring them to atmospheric air at the same temperature and under the same pressure as the gases them- selves. 31. Methods of Finding Specific Oravity.— The law of tlie buoyant effort, or upward pressure, of water can be made 'se of to determine the specific gravities of bodies; for, if a b pnd the speeifie gravity of a liquid. Take a solid whicli is si)e('ificiilly heavier than eitlier the Ii(jnid or vuter, and let it be weighed in both ; then the loss ot weight in the two cases will be the respective weights of equal volumes of the liquid and of water ; therefore, the toss of ujeight in the liquid, divided by the lof>s of weight in the water, will give the specific gravity of the liquid. Lc* w = the weight of the solid in air, w' = its weight in the liquid whose specific gravity is to be determined, and w^ = its weight in water; then w -— w' and w — lo^ are the respective weights of equal volumes of the liquid and of water; therefore H^'l^Ul. . (3) otherwise thus : Let w = the weight of an empty flask, io' = its weight when filled with the liquid, and Wj = its weight wheii filled with water ; then w' — ?/.' am. w^ — w are the respective weights of equal volumes of the liquid and of water ; therefore S = w — w (4) EXAMPLES. 1. A cubical iceberg is 100 ft. above the level of the sea, its sides In'ing vertical. iVwrnx the specific gravity of sea- water = 1.02<»3, and of ice = 0.9214 at the temperature of 32°, to find its dimensions. 78 SPECIFIC GRAVITY OF BROKEN SOLID. Let X = the length of one side, X — 100 = the length of the piece under water; then we have (Art. 25, Cor.), a^ : rea — 100a?» :: 1.0263 : 0.9214 ; .-. a; : 100 :: 1.0263 : 0.1049; .-. x = 978.3 ft., and a^ = 936,302,451.687 cu. ft. 2. A piece of limestone, whose weight is 256.34 lbs., weighs in water 159.13 lbs. Find its speci/ic gravity. Ans. 2.637. 3. Find the specific gravity of a jncce of cork whoso weight is 20 grains. To sink it, we attach a biiiss weight which, when immersed in the water, weighs 87.22 grains; the weight of the compound body wheVi immersed is 23.89 grains. Ans. 0.24. 4. A solid weigliin"—«;'= weight of solid pieces— wt. of water they displaco = w — weight of water disi)Uiced ; thereforo w 4- w' — w" = weight of water displaced ; w S = w + w' — w" (1) SPECIFIC QRA VITY OF A MIXTURE. 78 33. Specific Gravity of Air. — Take a large flask which can be completely closed by a stop-cock, and weigh it when filled with air ; withdraw the air by means of an air- pump and weigh the flask again ; finally, fill the flask with water and weigh again. This lust weight minus the second will give the weight of the water that filled the flask, and the first weight minus the second will give the weight of an equal volume of air; divide the weight of the air by that of the water ; the result will be the siwcific gravity of air as compared with that of water. Let IV = the weight of the exhausted flask; w', w" its weights when filled with air and water ; then weight of the air CMintained by the flask, weight of the water contained by the flask; w — 10 It to 10 therefore. H w — w w"— 10 (1) Sen. — In the same manner the specific gravity of any gas can be obtained. The specific gravity of water at 20°.5 is about 708 times that of air at 0° under the pressure of 1^9.9 inches of mercury at 0". The atmosphere in which these operations must be per- formed varies at different times, even during the same day, in respect to temperature, the weight of its column which presses upon the earth, j'ud the (juantity of moisture it con- tains. On these accounts, corrections must be made before the specific gravity of air, or thai of any gas exposed to its pressure, can be accurately determined. The discussion of the principles according to which these corrections are made, is given in Chap. III. 34. Specific (Jravity of a Mixture.— (1) mm the t'itl itnicfi and specific gravities of flic coin/miunts . Tlic Hydrostatic Balance. — In o.der to deter- mine the specific gravities of bodies practically and with accuracy, it is necessary to employ certain instrnments for weighing. These are the I{f/(lro,static Balanre and HydrometerH.* The hydrostatic balance is an ordi- nary balance, having one of the scale- pans smaller than the other, and at a less distance from the beam ; attached to the under side of the small scale- l)an is a hook, from which may be suspended nr.} body by means of a thin platinum wire, horse-hair, or any delicate thread. Tlie body whose S|)ecific gravity is to be found ig ,1- THE COM MO X HYDROMETER. 77 suspended from the hook, and then its weight i < determined. It is then weighed in water, and thus its los^ of weight is ascertained, which is the weight of a portion cf water equal in volume to the hody. 37. The Common Hydrometer.— Tiie lame hydrom- eter is given to a class of instruments used for determiniiig the specific gravities of liquids by observing either tlje depths to which tliey sink in the li(iuid8 or the weights re- quired to make them sink to a given dopth. Tiiese instruments depend upon the principle that the weight of a floating body is e(pial to the weight of the fluid which it displaces. The common hifdrome/er is usually nuu!- of glass, and consists of a straight stem ending in two hollow spheres, li and (> the lower one being loaded so as to keep tlic iiistrument in a vertical position when floating in the li(piid. There are no weights used with the instrument; but the stem is graduated, so as to enable the operator to ascer- tain the specific gravity of a li«|ui(i by the de[)th to which the instrument Hini\s in it. E D -:^ Fig. 27 Let k = the area of a seerio.i of the stem, r ■= the vol- ume, and w = the weight of tlie hydrometer. When the hydrometer fioats in a liquid wh ^se spe('ifi(r gravity is tt, let the level I) of the stem be in the surfaice ; and when it floats in a liipiid wiiost' specific gravity is s', let the level E be in the surface. Then (Art. .34, Rem.) we have for the weights of the liquid displacvd in the first and secondares, respeetively, w = .>(r — it- AD), v = .s' {v - /(-.AK) ; but the weight of the licjuid displaced in each case is the same, since eacli is etjual to the weight of the instrument. 78 SIKJiaS JI YDli OMKTKJi. V — k' AE /jAD' V w whicli gives the ratio of the specific gravities of the two liquids. • . C'oR.— If llio second liquid bo the standard, s' = 1, and .s, the specific gravity of the first li«piid, is given in (1). •{8. Sikos's Hydronu'ter.*— This instrument differs from the common hydnuiu'ter in (he shape of the stem, whicli is a flat bar and vecific grtivity of tho first litjuiil, ife ^iven in (1). 39. NicholNOirs Hydrometer.— The two hydrometers just described are used fur obtuiiiing the specitie gravities of h(|uids. NichohoHx hiidrometcr is ko contrived Jis to determine i\w s|)ecific gravity of solids as well as liquids. It consists of a hollow metidlic vessel C, gener- ally of bniss, terminated above by a very thin stem, which is often a steel wire, bearing a small dish A, and carrying at its lower end a heavy cuj) D; on tho stem connecting A and C, a well- defined mark B is nuide. h X (1) Ti) (U'lvnninf the sftcrip'r ^rni^itij of iv Uqiiiit. Let w be the weight of the hydrometer, w' the weight which must be placed in the dish A, in order to sink the stem to the point Ji in a iifjuid whose specific gniv'ty is *, and w" the weight which must be placed in the dish A, lu sink the stem to the same point \\ in a litjuid whose s))ecific gravity is s'. Then we have for the weights of the liquid displaced in the first and seccmd cases, resiKJCtively, w 4- w' and w -{- w" ; and since the volume* displaced are the sjime in both cases, the siK'cific gravities are as the weights (Art. 20), " 8' - w + ?<; n 0) Calling the second liquid the standard, «' = 1, and (1) becomes ." y w -I- w ivhich in the ^fjcci/ic gravity required. 80 EXAMPLES. (2) To (Irfrrt)n'nr the sprri/fr ^rantij of (v sofifJ. Let w Im) the weight which mii8t he phiocd in the dish A, to uink the Hterii to tiie point li in u liquid whuse speoitio gravity is «. Put the solid in the d>Hh A, and let w' be the weight which must he added to the 8olid to sink the stem to the point B in the same litpiid. . Then put the solid in tin* lower dish D, and let /r" he the weight re(|uired in the upper (lish A to sink the stem to the point \\ in the same liquid. {fence, the weight of the 8y the solid = w" — w/. Hence, denoting by iS the .s|)ecili(' gravity of the solid, we have s w — w w — w If the li((ui(l is the standard, x = 1, and (3) h(H!onu's .S' = w — w ivhif'h is the s/trri/ir ^raififi/ rrqiUred. EXAMPLES. 1. If an iceberg whose density is 0.9 IH float in n liquid whoso density is 1.028, what is the ratio of the part sub- ni('rg"d to that which is above water? Am. 8.3 : 1. 2. II(»w much of its weight will 112 lbs. of iron lose, if immersed in water, tho density of iron being 7.25 times that of water? ylw.v. 15.448 lbs. 3. If 20 lbs. of cork Ik; immersed in water, with what force will it rise towairds the surface, its density being 0.24 times that of water ? Am* 03^ lbs. EXAMPLES. 81 4. If a piece of wood whoge vertical height is 2 ft. l)e ])lu('e(l in the Dead Sea, huw many inches will it liocume 8iil)mergcd, the deuHitii>8 uf the wuod and Dead Sea water being .53 and 1.5J4 respeetively ? Ans, 10.2G ins. 5. Find the depth to which a rectangular hloek will sink in water, the depth of the block being a feet, and the weight of each cubic foot of it being w lbs. , aw A us. (iJi.6 C. A barge of a rectan^jnlar shape is I ft. long, /> ft. brojul, and a ft. deep, out^^ide measure. The thickness of the planking is c ft., and the weight of a cubic foot of the tim- ber is w lbs. To what depth will the barge sink when loaded with W lbs. ? 7. A cylindrical piece of wood, weight IT, floats in water witii its axis vertical and immersed t^» a depth h. Find how much it will be dei>reBsed by placing a weight w on the top of it. M w , Ans. j.rh. W 8. An isosceles triangle floats in water with its base hori- zontal. Find the ])osition of e(|uilibrium when the base is above the surface, its height being h and its density being I that of wat^r. Ans. irVC. 0. A rectangular barge, / ft. long, h ft. broad, and a ft. deep, (mtsidc measure, sinks to \ its whole depth wlioii un- loaded, liequired its weight in lbs. An». VZ.hahl. 10. If a rectangular barge sinks to \ of its whole depth when unloaded, and to J of its whole depth when loaded, find the load, the weight of the barge being to. Ans. \w. 11. The diameter of the base of a right cone is 2r, its altitude is h, and its density is f that of water. To what IMAGE EVALUATION TEST TARGET (MT-3) % i.O 1.1 L4 12.8 i^ M3. 12.0 lit i 1.8 1 1.25 |,,.4 ,,.6 < 6" ► Photographic Sdences Corporation 23 WIST MAIN SYMiT WiBSTEII,N.Y. 14S«<« (716) •72-4503 >^* <* ^4^ dp 83 EXAMPLES. depth will the cone sink when it floats with its vertex down- wards ? A ■ '* 8/75 Ans. ;rVl8. o 12. A hemispherical vessel, whose weight is w, floats upon a fluid with ^ of its radius below the surface. What weight must be put into the vessel so that it may float with f of its radius below the surface ? Ans. fw. 13. Let the pontoon in Ex. 4, Art. 26, be a cylinder, length /, with liemispherical ends, radius r; to find the load requisite to sink the pontoon to a given depth u. Ans. [Al + 7ra2(r — ^)] 62.5, where A = the area ADK (Fig. 22). 14. Required the thickness of a hollow globe of copper whose density is 9 times that of Avater, so that it may just float when wholly immersed in water, r being the exterior radius. Ans. r(l - |\/3). 15. A cubical box, the volume of which is one cubic foot, is three-fourths filled with water, and a leaden ball, the volume of which is 72 cubic inches, is lowered into the water by a string. It is required to find the increase of pressure (1) on the base and (2) on a side of the box. . Ans. (l)41foz.; (2) 32+ oz. 16. If the height of the parallelepiped in Ex. 2, Art. 28, is 0.9 of the breadth, and if p = |, find the inclination that the parallelepiped may float in indifferent equilibrium. Ans. e = 33° 15'. \\. What is the weight of a cube of gold whose side is 3 ins., its specific gravity being 19.35 ? Ans. 18.890 lbs. 18. What is the volume of a piece of platinum whose weight is 10 lbs., its specific gravity being 22.06 ? Ans. 12.533 cu. ins. 19. A piece of lead, whose weight is 511.65 grs., weighs in water 466.57 grs. Required its specific gravity. Ans. 11.35. EXAMPLES. 83 20. A sovereign, whose weight is 123.02 grs., weighs in water 116.02 grs. Required its specific gravity. Ans. 17.574. 21. Find the specific gravity of a piece of wood whose weight is 50 grs. To sink it we attach a brass weight which, when immersed in the water, weighs 87.22 grs. ; the weight of the compound body when immersed is 42.88. Ans. 0.53. 22. A piece of wood weighs 4 lbs. in air and a piece of lead weighs 4 lbs. in water ; the lead and wood together weigh 3 lbs. in water. 'Find the specific gravity of the wood. Ans. 0.8. 23. A body immersed in water is balanced by a weight P, to which it is attached by a string passing over a fixed pulley ; wheh half immersed, it is balanced in the same way by a weight 2P. Find the specific gravity of the body. Ans. f. 24. Find the weight of a cubical block of stone whose side is 4 ft., and specific gravity 1|. Ans. 80000 oz. 25. A body weighing 20 grs. has a specific gravity of 2\. Required its weight in water. Ans. 12 grs. 26. An island of ice rises 30 ft. out of the water, and its upper surface contains f of an acre. Supposing the mass to be cylindrical, required (1) its weight, and (2) depth below the water, the specific gravity of sea-water being 1.0263, and that of ice .92. Ans. (1) 242900 tons; ^2) 259.64 ft. 27. A piece of wood weighs 12 lbs., and when attached to 22 lbs. of lead and immersed in water, the whole weighs 8 lbs. The specific gravity of lead being 11, required that of the wood. Ans. \. 28. A solid which is lighter than water weighs 5 lbs., and when it is attached to a piece of metal, the whole weighs 7 lbs. in water. The weight of the metal in water u EXAMPLES. being 9 lbs., compare the specific gravities of the solid and of water. Atis. 5 : 7. 39. A piece of wood which weighs 57 lbs. in vacuo, is at- tached to a bar of silver weighing 42 lbs., and the two together weigh 38 lbs. in water. Find the specific gravity of the wood, that of water being 1, and that of silver 10.5. Atis. 1. 30. Equal weights of two fluids, whose specific gravities are s and 2s, are mixed together, and one-third of the whole volume is lost. Find the specific gravity of the resulting fluid. Ans. 2s. 31. Tv/o fluids of equal volume, ajid of specific gravities s and 2s, lose I of their whole volume when mixed together. Find the specific gravity of the mixture. Ans. 2s, 32. A cylinder floats vertically in a fluid with 8 ft. of its length above the fluid ; find the whole length of the cylin- der, the specific gravity of the fluid being three times that of the cylinder. Ans. 12 ft. 33. A body floats in one fluid with | of its volume im- mersed, and in another with | immersed. Compare the specific gravities of the two fluids. Ans, 15 : 16. 34. A block of wood, the volume of which is 4 cubic feet, floats half immer^isd in water. Find the volume of a piece of metal, the specific gravity of which is 7 times that of the wood, which, when attached to the lower portion of the wood, will just cause it to sink. Ans. | of a cubic foot. 35. A cone, whose specific gravity is ^, floats on the water with its axis vertical, (1) with its vertex downwards and (2) with its vertex upwards. What part of the axis is immersed in each case? - Ans. (1) i ; (2)0.0436. 36. A cone, whose specific gravity is ^, floats with its axis vertical. Compare the portions of the axis immersed, (1) when the vertex is upwards, (2) when it is downwards. Ans. ^'2-1:1. EXAMPLES. 85 37. A block of ice, the volume of which is a cubic yard, Is observed to float with -^ of its volume above the surface, and a small piece of granite is seen embedded in the ice. Find the size of the stone, the specific gravities of ice and granite being respectively .918 and 2.65. Ans. ^1^ of a yard. 38. A cylindrical glass cup weighs 8 ozs., its external radius is 1^ ins., and its height 4|^ ins. If it be allowed to float in water with its axis vertical, find what additional weight must be placed in it, in order that it may sink. -8) Ans, I- 64 oz. 39. Find the position of equilibrium of a cone, floating with its axis vertical and vertex upwards, in a fluid of which the density bears to the density of the cone the ratio 27 : 19. ns. \ of the axis is immersed. 40. The whole volume of a hydrometer is 5 cu. ins., and its stem is \ of an inch in diameter; the hydrometer floats in a liquid A, with one incli of the stem above the surface, and in a liquid B with two ins. above the surface. Compare the specific gravities of A and B. Ans. 1280 — TT : 1280 — 27r. 41. What volume of cork, specific gravity .24, must be attached to 6 lbs. of iron, specific gi'avity 7.6, in order that the whole may just float in water ? -^^*'' Ti^ of ^ cubic foot. 42. If a piece of metal weigh in vacuum 200 grs. more than in water, and 160 grs. more than in spirit, what is the specific gravity of spirit ? Ans. ^. 43. A piece of metal whose weight in water is 15 ozs., is attached to a piece of wood, which weighs 20 ozs. in vacuum, and the weight of the two in water is 10 ozs. Find the spe- cific gravity of the wood. Ana, \. 86 EXAMPLES. 44. A crystal of salt weighs 6.3 grs. in air; when covered with wax, the specific gravity of which is .9(3, the whole weighs 8.22 grs. in air and 3.02 in water. Find the specific gravity of salt. Ans. 1.9 nearly. 45. A Nicholson's Hydrometer v/eighs 6 ozs., and it is requisite to place weights of 1 oz. and 1^ ozs. in the upper cup to sink the instrument to the same point in two differ- ent liquids. Compare the specific gravities of the liquids. Ans.'4:\b. 40. A diamond ring weighs 69|^ grs., and G4| grs. in water. The specific gravity of gold being 16|, and that of diamond 3|, what is the weight of tlie diamond ? Ans. 3f grs. 47. A body A weighs 10 grs. in water, and a body B weighs 14 grs. in air, and A and B together weigh 7 grs. in water. The specific gravity of air being .0013, required (1) the specific gravity of B, and (2) the number of grs. of water equal to it in volume. Ans, (1) .8237; (2) 17.023 grs. 48. A compound of gold and silver, weighing 10 lbs., has a specific gravity of 14, that of gold, being 19.3, and that of silver being 10.5. Required the weights of the gold and the silver in the compound. Ans. Gold = 5.483 lbs.; silver = 4.517 lbs. 49. A diamond ring weighs 65 grs. in air and 60 in water. Find the weight of the diamond, if the specific gravity of gold is 17.5, and that of the diamond 3|. Ans. 6.875 grs. 50. The crown made for Hiero, King of Syracuse (Art. 24, note), with equal weights of gold and silver, were all weighed in water ; the crown lost ^ of its weight, the gold lost T^ of its weight, and the silver lost -^ of its weight. Prove that the gold and silver were mixed in the proportion of 11:9. ► 1 EXAMPLES. 87 61. A ring consists of gold, a diamond, and two equal rubies ; it weighs 44^- grs., and in water 38| grs. ; when one ruby is taken out, it weighs 3 grs. less in water. Find the weight of the diamond, the specific gravity of gold being 16J, of diamond 3^, of ruby 3. Ans, b\ grs. 52. If the price of pure whiskey be $4 per gallon, and its specific gravity be .75, what should be the price of a mixture of whiskey and water which on gauging is found to be of specific gravity .8? Ans, $3.20. 53. How deep will a paraboloid sink in a fluid whose spe- cific gravity is n times that of the solid, the axis being vertical and equal to a, and the vertex upwards ? , Vw — Vw — 1 Ans. a- V n 54. A cubic inch of metal, whose specific gravity is m, is formed into a hollow cone, and immersed with its vertex downwards. Determine the ratio of the altitude to the exterior radius of its base, when the surface immersed is a minimum. Ans. V^. CHAPTER III. EQUILIBRIUM AND PRESSURE OF GASES. — ELASTIC FLUIDS. 40. Elasticity of Gases.— The pressure of an elastic fluid is measured exactly in the same way as the pressure of a liquid (Art. 6), and the equality of pressure in every direc- tion, and of transmission of i)ressure, are equally true of liquids and gases (Arts. 7 and 8). There is, however, this difference between a liquid and a gas : when a liquid is con- fined in a vessel, no pressure is exerted against the sides except that which is due to the weight of the liquid itself, or that which is transmitted by tlie liquid from some point on the surface at which, an external force is applied; whereas, if a gas be contained in a closed vessel, there is, although modified by the action of gravity, an outward pressure exerted against the sides, which is due to the elas- ticity of the gas, and which depends upon its volume and temperature.* It is therefore evident that genemlly a gas cannot have a free surface like a liquid (Art. 11), for such a surface implies that at each point the pressure is nothing, i. e., if it be covered by an envelope everywhere in close con- tact with it, no pressure is exerted against the envelope. It is also evident that, if a portion of the gas be withdrawn from the vessel, that which remains will not fill the same part of the vessel that it occupied before, as in the case of a liquid, but will expand so as to fill the whole vessel, press- ing, but with diminished force, against its sides at every point (Art. 2). From this property of gases, they dr*^ called elastic fluids; the outward pressure which a gas exerts t * If the gae is not confined within a limited space, the effect of its elasticity tnlgbt be the anlimited ezpaaeion and nltimate dispersion of the gas. PRESSURE OF THE ATMOSPHERE. 80 I against the walls of the vessel enclosing it is called its elas- lie force. The action of a common syringe will serve to illustrate the elasticity of atnio8i)heric air. If the piston be drawn out, and the open end of the syringe then closed, a consid- erable effort will be required to force in the piston to more than a small part of the length of its range, and if the feyringe be air-tight and strong enough, it will require the application of great power to force the piston down through nearly the whole of its range. This experiment also shows that the pressure increases with the compression, the air within the syringe acting as an elastic cushion. If the piston be let go, after being forced in, it will be driven back, the air within expanding to its original volume. An inverted glass cylinder, carefully immersed in water, funiishcs another simple illustration of the elasticity of air. Holding the cylinder vertical, it may be pressed down in the water without much loss of air, and it will be seen that the sur- face of the water within the vessel CD is below the surface of the water outside AB. It is evident that the downward pressure of the air within at CD is equal to the upward pressure of the water at the same place, which (Art. 11, Cor. 2) is equal to the pressure on the up- per surface AB, increased by the pressure due to the depth of the surface CD below the upper surface ; hence the air within, which has a diminished volume, has an increased pressure. 41. Pressure of the Atmosphere. — If a glass tube* about three feet in length, closed at one end, be filled with mercury, and then, with the finger pressed to the open end * This experiment was first made by Torricelli, and hence is called TorriceUi's Experiment, and the vacant space above the mercury in the tube is called the Toni- ceUian Vacuum, A B .=-_— C D ;jr_~~ " ■l^^r=T- Fig. 30 90 WEIGHT OF THE AIR. SO as to close it, inverted in a vessel of mercury so as to im- merse its open end, it will be found on removing the finger that the mercury in the tube will descend through a certain 8])ace, leaving a vacuum at the top of the tube, but resting with its ui)per surface at a heiglit of about 29 or 30 inchcM above the surface of the mercurv in the vessel. It thus appears that the atmospheric pressure, acting on the sui-- face of the mercury in the vessel, and transmitted (Art. 8)j supports the column of mercury in the tube, and hence that the weight of the mercurial column is exactly equal to the weight of the atmospheric column standing on an area equal to that of the internal section of the tube. The weight of this column of mercury then is an exact measure of the atmospheric pressure, or of the elastic force of the atmosphere at any instant. 42, Weight of the Air. — This may be directly proved by weighing a flask filled with air, and afterwards weighing it when the air has been withdrawn by means of an air- pump ; the difference of the weights is the weight of the air contained by the flask. The opinion was long held that air was without weight, or rather, it never occurred to any of the piiilosophers who preceded Galileo to attribute any influence in natural phe- nomena to the weight of the air. The fact that air has weight escapes common observation in consequence of its extreme levity compared with solids and liquids, and espe- cially in consequence of its being the medium by which we are continually surrounded. The experiment of weighing air was performed successfully for the first time in 1650, by Otto Guericke, the inventor of the air-pump.* By means of the weight of air we may account for the fact of atmospheric pressure. The earth is surrounded by a quantity of air, the height of which is limited (see Art. . * DeBCbftners NAtoral Philoeophy, P&rt I., p. 141, THE BAROMETER. 91 5 72) ; and if we suppose a cylindrical column extending above any horizontal area to the surface of the atmosphere, the weight of the column of air must be entirely supported by the horizontal area upon which it rests, and the pressure upon the area is therefore equal to the weight of the column of air. The pressure of the air must then diminish as tiie iieight above the earth's surface increases ; and from exper- iments in balloons and in mountain ascents, this is found to be the case. The action of gravity is equivalent to the effect of a compression of the gas, and it is thus seen that the pressure of a gas is in fact caused by its weight, as in the case of a liquid. Taking tt for the pressure of the air at any given place (Art. 11, Cor. 2), and assuming that the density of tiie air throughout the height z is constant and equal to p, the pressure at the height z will be -n — gpz. (1) Cor. — It may be shown, in the same manner as for air, that any other gas has weight, and that the intrinsic weight is in general diflferent for different gases. Carbonic acid gas, for instance, is heavier than air, and this is illustrated by the fact that it can be poured, as if it were liquid, from one jar to another. 43. The Barometer. — This instrument, wliich is employed for measuring the pressure of the atmosphere, is, in its simplest form, a straight glass tube AB, about 32 or 33 inches long, containing mercury, and having its lower end immersed in a small cistern of mercury ; the end A is hermetically sealed, and there is no air in the branch AB. Since the pressure of a fluid at rest is the same at all points of the same horizontal plane (Art. 10), the pressure at B, in the interior of the tube, is equal to B Fig. 31 92 THE WATER-BABOMETEE, the atmospheric pressure on the mercury at C, which is transmitted from the surface of tlie mercury in the cistern to the interior of the tube; and as there is no pressure on the surface at P, it is clear that the pressure of the air on is the force which sustains the cohimn of mercury PB. Let a be the density of mercury, and n the atmospheric pressure at C ; then we have 7r = r/rTPB, (1) and, since g and a are constant, the height PB may bo used as a measure of the atmospheric pressure. 44. The Mean Barometric Height. — The mean lieiglit of the barometric column at the level of the sea is found to vaiy with the latitude, but it is generally between 29^ and 30 inches. The atmosphere is subject to continual changes, some irregular, others periodical. If the density and consequent clastic force of the air be increased, the col- umn of mercury will rise till it reaches a corresponding increase of weight ; if, on the contrary, the density of the air diminish, the column will fall till its diminished Aveight is sufficient to restore the equilibrium. The barometric height is therefore subject to continuous variations ; during any one day there is an oscillation in the column, and the mean height for one day is itself subject to an annual oscil- lation, independently of irregular and rapid oscillations due to high winds and stormy weather. Usually the height of the column is a maximum about 9 A. m. ; it then descends until 3 p. M., and again attains a maximum at 9 p.m.* 45. The Water-Barometer. — Mercury possesses two great advantages over other liquids, which has led to its being selected above all others for use in barometric instru- ments. The first advantage of mercury is that it does not give off vapor at ordinary temperatures. If it did, the space .. * Besant'e HydroBtatics, p. 76. MANOMETERS, AP above the mercury would be tilled with an elastic vapor, which would press down upon the column, so that its weight would no longer be a measure cf the atmospheric pressure, but of the difference between this pressure and the elastic force of tlie vapor given off. The second advantage is that, on account of the great density of mercury, the height of the column which measures the atmospheric pressure is so small that barometers constructed with it are of a very con- venient size. Tiie pressure of the air may be measured by using any kind of liquid. The density of mercury is about 13.595 times that of water,* and therefore, if water were used, it would be necessary to have a tube of great length, since the column of water in the water-barometer would be about 33f feet. In order to measure easily and correctly the barometric height, an accurately graduated scale is added, which can be moved along the tube. Rem. — The instrument above described involves the essen- tial parts of a barometer ; it is the province of Physics to give a full description of different kinds of barometers, to explain their use, etc. 46. Manometers. — Barometers are used not only to measure the pressure of the external air, but also to deter- mine the elastic force of gases or vapors which are enclosed in vessels. When thus used, they are called manometers. Thcso instruments are filled with mercury, and are either open or closed ; in the latter case, there may be air above the column of mercury Or there may be a vacuum. The manom- eter with a vacuum above the column of mercury is like the common barometer. In order to measure with it the elastic force of the gas or vapor, it will be necessary to establish a free connection between the cistern of the barometer and the vessel containing the fluid. This is done by means of a .. . ♦ Enc. Brit., Vol. XVI, p. 88. d4 THE ATMOSPHERIC PRESSURE. ■30 fpi g.32 tube DE, one end of which E opens into the vessel contain- ing the fluid, and the other end D enters above the level of the mercury B in the cistern. By this means the gas from the vessel flows through the tube ED into the cistern, and presses a column of mercury into the tube AB, the height of which measures the elastic force of the gas or vapor in the vessel. When the elastic force of the fluid is consid- erable, it is usual to estimate it as so many atmospheres : for instance, steam, in the boiler of an engine, having a pressure of two atmos- pheres, signifies that its elastic force would sus- tain a column of about 60 inches of mercury. If it is said to have a pressure of 6 atmospheres, it means that its elastic force would sustain a column of about 180 inches of mercury ; and so on. 47. The Atmospheric Pressure on a Square Inch. — This may be found at once by observing that it is the weight of a cylindrical column of mercury whose base is a square inch, and whose height is equal to that of the barometric column. Since the specific gravity of mercury is 13.595, that of water being 1, it follows that the pressure of the air on a square inch, taking 30 inches as the height of the barometer at the sea level, = (30 X 13.595 X 62.5-^-1728) lbs. = 14.7 lbs., and this is called the pressure of one atmosphere. ScH. — This pressure varies from time to time, but is gen- erally between 14| and 15 lbs. The standard usually adopted where the English system of measure is used is 14.7 lbs. upon the square inch, which corresponds to a col- BOYLE AND MARIOTTB'S LAW. 9$ ' umn of mercury about 30 (exactly 29.922) inches, and to a column of water about 34 (exactly 33.9) feet high. A press- ure of two atmospheres, therefore, would mean a pressure of 29.4 lbs. on each square inch, and a pressure of six atmos- pheres would mean a pressure of 88.2 lbs. on each square inch. (See Weisbach's Mechs., p. 777.) EXAMPLES. 1. If the elastic force of a gas is 2^ atmospheres, find its pressure in lbs. on each sqi.are inch. Ans. 36.75 lbs. 2. If the elastic force of steam in a boiler be 5^ atmos- pheres, find the pressure on a safety-valve whose area is 5.4 sq. ins. Ans. 436.59 lbs. 48. Boyle and Mariotte's Law.*— Gases readilv con- tract into smaller volumes when compressed. "When a gas is compressed, its elastic force is increased ; and when it is allowed to expand, its elastic force is diminished. The statement of the law which expresses the relation between the pressure and the volume, or the pressure and the density, of gases is the following : The pvessiivc of a given quantHy of air, p^^ nt a given temperature, varies inversely as its volume, and directly as its density. M H-" E N -K F Let ABCD be a bent glass tube, the shorter branch of which can have its end D closed, and both branches being vertical. Let a little mercury be poured in at A, and let it stand at the same level EF in both branches. Now close the end D ; a definite volume of air is thus enclosed in DE under a pressure equal to that of the external air, i. e., the elastic force of the enclosed air DE is equal to the atmospheric pressure exerted on F in B Fig. 33 * The experimental proof of thie law was discovered about the eame time in England by the Hon, Robert Boyle, &nd in France by Mariotte, 96 BOYLE AND MARIOTTE'S LAW. i the open branch, and is therefore equal to one atmosphere (Art. 47). Take DH = ^DE, and pour mercury slowly into the tube AB till it stands at H in the shorter branch ; in the longer branch it will be found to stand at the height LK = 30 inches above HK, i.e., the me/cury, rising in the shorter branch, compresses the air which it drives before it, and when the air in the shorter branch is reduced to half its volume, its elastic force or pressure is two atmospheres, since it now sustains not only the atmospheric pressure which is exerted on the surface of the mercury in the open branch, but also the weight of a column of mercury 30 inches high. When mercury is poured into the tube till it rises in the shorter branch to M, where DM = ^DE, it will be found to stand in the longer branch at the height AN = 60 inches above MN, i. e., when the air in the shorter branch is reduced to one-third of its volume, its elastic force or pressure is three atmospheres, since it now sustains the atmospheric pressure and the weight of a column of mercury 60 inches in height. In the same way, it may be shown that if the air occupy one-fourth of its original vol- ume DE, it will sustain a pressure of four atmospheres, and so on for any number. Hence, generally, the pressure of a quantity of air varies inversely as ifs volume. When the volume is reduced to one-half, the density is doubled ; Avhen reduced to one-third, the density is trebled, and so on ; that is, the volume varies inversely as the density* Hence, the pressure varies directly as the density. Let V and v' be the volumes of a given mass of air, p and p' the corresponding pressures, and p and p' the correspond- ing densities. Then we have J > p _ V p = Icp, where ^ is a constant to be determined by experiment. (1) (») \ i EXAMPLES. 97 Rem. 1. — It has been shown by a seres of experiments that this law connecting the elastic force and volume of a gas under a constant temperature is sensibly true for air and most gases as far as a pressure of 100 atmospheres.* It is only when the pressures are very great that variations from the law are observed, and even then the departure from the law is but small, especially with those gases which we are not able to condense into liquids. "With gases wiiich undergo liquefaction at moderate pressures, the departure from the law is greater, and increases as the state of liquefaction is approached, f Rem. 2. — In conducting this experiment, care must be taken to have the temperatures the same at the beginning and at the conclusion, as the elastic force of a gas under a given volume is influenced by changes of temperature. For this reason, it is necessary to pour in the mercury gradually, and to allow some time to eli".pse before the difference of levels is observed, since, whenever a gas is compressed, an elevation of temperature is produced. Therefore, whatever heat is developed by increase of pressure must be allowed to pass off before the volume of gas is observed. \ EXAMPLES. 1. Let DE (Fig. 33), be 10 inches ; if mercury be poured in until the level in the closed branch stands 3 inches above EF, and in the open branch 15.64 inches, find the elastic force of the air in the closed branch, the barometer standing at 29.5 inches. Since the levels of the mercury in the two branches stand at 15.64 and 3 inches, the level in the longer branch is 12.64 inches above that in the closed branch; the elastic force of the compressed air, therefore, sustains a column of * Oalbraith's HydroBtatics, p. 35. t Weisbacb'B Mechs., p. 783 ; also Twiaden's Mechs., p. 299. 08 EFFECT OF HEAT ON GASES. mercury 12.64 inches high, together with the atmospheric pressure, which by the barometer is shown to be equal to a column of 39.5 inches ; hence the elastic force = (12.64 4- 29.5) inches =-■ 42.14 inches. 2. If the level in the closed branch rise 6.4 inches, find the height to which the level in the open branch should rise, the barometer standing at 30.42 inches, and DE being 10 inches. Ans. 60.48 inches. 49. Eifect of Heat on (iases.— When a given quan- tity of air or gas is increased in temperature, it is found that, if the air or gas cannot change its volume, its elastic force is increased ; but if the air can expand freely, while its elastic force remains the same, its volume will be in- creased. To illustrate this, take an ai^-tight piston in a vertical cylinder containing air, and let it be in equilibrium, the weight of the piston being supported by the cushion of air beneath it. Raise the temperature of the air in the cylinder by mimersing it in hot water; (1) the piston will rise in the cylinder as the volume of the heated air expands ; and when the air has reached the temperature of the surrounding water, tlic piston will cease to ascend, and will remain sta- tionary. But (2) if we suppose that when the heat is applied, the piston is held down so as to keep the air under a constant volume, an effort will be required to prevent the piston from ascending in the tube, which becomes greater in proportion as the air is heated. Hence (1) The effect of heat on a given quantity of air, the elastic force remaining constant, is to expand its vohnne. (2) TJte eff'ect of heat on a given quantity of air, the volume remaining constant, is to increase its elastic force. .... I f / THERMOMETERS. 99 50. Thermometers.— As a general rule, bodies expand under the action of heat, and contract under the action of cold, and the only method of measuring temperatures is by observing the extent of the expansion or contraction of some known substance. Any body which indicates changes of temperature may be called a thermometer. As the expansions of different substances are not exactly proportional to one another, it is necessary to select some one substance or combination of substances to furnish a standard, and the standard usually adopted for all ordinary temperatures is the apparent expansion of mercury in a graduated glass vessel ; for very high temperatures, a metal of some kind is the more useful, and for very low tempera- tures, at which mercury freezes, alcohol must be employed. The mercurial thermometer is formed of a thin glass tube of uniform bore, terminating in a bulb, and having its upper end hermetically sealed. The bulb contains mercury, which also extends partly up the tube, and the space between the mercury and the top of the tube is a vacuum. Since the glass, as well as the mercury, expands with jin increase of temperature, the apparent expansion is the difference be- tween the actual expansion and the expansion of the glass. The construction of an accurate mercurial thermometer is an operation of great delicacy. In Fahrenheifs Thermometer, which is chiefly used in England and in this country, the freezing point is marked 32°, and the boiling point 212". The space, therefore, be- tween these two points is 180°. In the Centigrade Thermometer the freezing point is marked 0°, and the boiling point 100°, the space between being divided into 100°. In Reaumur's Thermometer the freezing point is also marked 0°, but the boiling point is marked 80° *. * The temperatare indicated by the boillDg point is the eame in all. 100 EXAMPLES. Rem.— Mercury freezes at a temperature of —40° C. or F., and boils at a temperature of about 350" C. or 663' F. ; it is therefore necessary, for very high or very low temperatures, to employ other substances. For very low temperatures, spirit of wine is used ; this liquid has never congealed, although a temperature of —140' C. has been ob- served, which is the lowest temperature yet attained.* High temperatures are compared by observing the expansion of bars o' metal or other solid substances, and instruments called pyrometers have been constructed for this purpose. 51. Comparison of the Scales of these Thermom- eters. — Any degrees of temperature by either thermometer may be converted into the corresponding degrees of the other thermometers ; for the space between the fixed points in Fahrenheit's being 180°, in the Centigrade 100°, .nd in Reaumur's 80°, we have 180° Fahrenheit = 100° Centi- grade = 80° Reaumur ; and therefore each of Fahrenheit's degrees = | of one of Centigrade = |^ of one of Reaumur. Let F, C, and R be the numbers of degrees marking the same temperature on the respective thermometers ; then since tlie space between the boiling and freezing points must in each case be divided in the same proportion by the mark of any given temperature, we iimst have Fj-S2 _ _ R_^ 180 ~ loo ~ 80 ' F-d2 C R . or 5 4 Rem. — The various scales were formed in the early part of the 18th century — Fahrenheit's in 1714, at Dantzic ; Reaumur's in 1731 ; and the Centigrade somewhat later.f EXAMPLES. 1. What temperatures on the other two scales are equiva- lent to the temperature 50° F. ? J Ans. 10° C, or 8° R. ♦ Maxwell on Heat. t Beeant's Hydroetatics, p. 88. X It is usual, in stating temperatures, to indicate the scale referred to by the ini- tials F., C, B. 1 D ALTON'S AND OAY-LUSSAC'S LAW. 101 « 2. Find (1) what temperature C. is the same as 60° R, and (2) what temperature R. is the same as 45° C. Arts. (1) 75° C. ; (2) 36° R, 62. Expansion of Mercury.— The expansion of mer- cury is very nearly uniform between 0° and 300°. Experi- ments show that, for an increase of 1° Centigrade, the expansion of mercury is j^j^, or .0001815 of its vohime ;* hence, if a< be the density at a temperature t, and ff,, the density at a temperature 0°, we have - a^ = at{i +.000180180; or, if we put .00018018 = 6, we have a, = *T, (1 + et), (1) which, in (1) of Art. 43, gives rr = gat PB = ga, (1 - et) PB, (2) by means of which the atmospheric pressure at any place can be calculated. 53. Dalton's and Gay-Lnssac's Law of the Ex- pansion of Gases by Heat. — The Following experimental law was discovered by Gay-Lussac f and Dalton, and more recently corrected by Regnault. // the j)vcssnrc reinaiiis coustaut, fin increase of temperature of 1° C. produces in a given mass of air an expansion of .003665 of its volume. By means of this experimental law, combined with Boyle's (Art. 48), the relation between the pressure, density, and temperature of a given mass of air or gas may be expressed. Conceive that a mass of air at the temperature of 0° C. is inclosed in a cylinder by a piston to which a given force is ♦ Enc. Brit., Vol. XVI., p. t See Pcecbaners Nat. Phil., p. a07. 102 DALTON'S A.\D GAIT-LUSSAC'S LAW. applied; let the temperature be increased to/; the piston will then be forced out until the original volume v^ is in- creased by .003665/^0, where v^ is the volume of air at 0°. Let V be the volume of the same mass of air at the tem- perature / ; then we have V =zVq{1 + .003665/) ; or, denoting .003665 by «, we have v = Vo(l 4- at). (1) Cor. 1.— If Fahrenheit's scale is used, the number of de- grees above the freezing point is / — 32 ; and, since 180" F. correspond to 100° C, the exi)ansion for 1° F. is — -- = 180 ^^2 of the volume at 32° F. The more accurate value of the denominator is 491.13. Hence, the increase of volume = -" ^ ~ ' ? 492 V and, for the whole volume, we have V = Vo + i'o(/-32) 492 or. 460 + t ' = ^'» -4-92 (2) where t is the temperature on Fahrenheit's scale, and v^ is the volume at 32° F. Cor. 2.— If v' be the volume which the same mass of air assumes at the temperature t', we have 460 + t' Dividing (3) by (3), we have , 460 + t' (3) W f PRESSURE^ TEMPERATURE, AND DENSITY. 103 I I i t I By means of (4) we may determine the volume which a gas will assume at a given temperature ; or, conversely, the temperature it will have under a given volume, if the volume it has at any given temperature is known, the pressure re- maining constant. EXAMPLES. 1. If 100 cubic inches of gas at 68° F. be heated to 130° F., find the volume, the pressure being constant. Ans. 109.85 cu. ins. 3. A mass of uir at 50° F. is raised to 51° F. What is the increase of its volume under a constant pressure ? i . , Ans. ^fy of its volume. 54. Law of the Pressure, Temperature, and Density of a Mass of Gas. — Let p, p, and v be the pressure, density, and volume of a mass of gas at the tem- perature t, vo and po the volume and density at 0°. Then, when p remains constant, we have, from (1) of Art. 53, v = v^{l-{- at). (1) Now, if t remains constant while the gas is compressed from V to <;„, the volume varies inversely as the density (Boyld's Law) ; that is. V : Vq :: Po ' P> (2) which in (1) gives, Po = P (1 + «0- Substituting in (2) of Art. 48, we have p = kp^ = 1cp{l -{-at). . (3) Cor. 1. — If />', p' be the pressure and density of the same gas at a temperature t', we have w (5) 104 PRESSURE, TEMPERATURE, AND DENSITY. P _ p 1 + ttt Cor. 2. — If the volume, and therefore the density, re- mains constant, while the temperature rises, the pressure will also rise. Let p^ be the pressure when ^ = 0, v and p remaining constant. Then (3) becomes, p^ = kp. Substituting in (3), we have ;> = jo„ (1 -f at), (6) where p and p^ are the pressures at the temperatures t and 0, the volume being constant. Let t =z 1, then (6) becomes p —pQ = p^a = .003665jOo (Art. 1)3) ; that is, // the volume of a mass of gas retnains con- stant, an increase of temperature of 1° 0. produces an increase of pressure equal to .003665 of its original pressure. C OR. 3. — If Fahrenheit's scale is used, (3), (4), and (6) become respectively , 460 -H t p^ _ p 460_-M p' ~ p' 460 + t 460 -f t n p =PI 493 (7) (8) (9) Cor. 4. — If jt>' be the pressure of the same gas at a temper- ature t', the volume remaining constant, we have, from (9), 460 + t' P =Pi 492 I . ABSOLUTE TEMPERATURE. •'• 7 ~ 460 + t" 105 (10) in which p and j»' are the pressures corresponding to the temperatures t and /' of a given mass of gas, the vohime being constant. Cor. 5. — Gince the volume of a given mass of air varies inversely as its density, we have, from (4) and (8), , \-\- at' p 1 -\- at p I } 460 -ht' p ^=-46o-:mV' (11) (12) where v' and v denote the volumes of a given mass of air at the temperatures t' and /. EXAMPL.ES. 1. If the pressure of a given mass of gas be 29.25 inches, at the temperature 56° F., what will it become if heated to 300° F., the volume being constant? Ans. 43.081 inches. 2. If 200 cubic inches of gas at 60° F. , under a pressure of 30 inches of mercury, be raised in temperature to 280° F., while the pressure is reduced to 20 inches, find the volume. Ans. 426.9 cubic inches. 55. Absolute Temperature.— If we can imagine the temperature of a gas lowered until its pressure vanishes, without any change of volume, we arrive at what is called the absolute zero of temperature, and absolute temperature is measured from this point.* Let /j represent this temperature on the Centigrade scale ; then (3) of Art. 54 becomes * BeMnt*B HjdromecbankB, p. 118. 106 or. PRESSURE OF A MIXTURE OF OASES. = kp (1 + (cU)^ 1 (1) U = = -JJ73'. In Fahrenheit's scale, the reading for absolute zero is -459°. Combining (1) of this Art. with (3) of Art. 54, we have p= kpa (/ — /J = kpa (/ + 273) = kpaT, {)>) where T is the absolute temperature. If V and p be the volume and density of a mass of gas, pv is constant, and therefore, from (2), ^ is constant ; from which it appears that the product of the pressure and volume of a giveu mass of gas is propoHional to the absolute temperature. ScH.— If the difference of temperature between the freez- ing and boiling points be divided into a hundred degrees, as in the Centigrade thermometer, the freezing point will then be 273° and the boiling point 373° absolute tempera- ture, and the zero of the scale will be that temperature at which the pressure vanishes. Denoting the absolute tem- perature by T, and the ordinary Centigrade temperature by /, we have T= 273° + ^. (3) 56. The Pressure of a Mixture of Gases.— If two liquids, which do not act chemically on each other, are mixed together in a vessel which remains at rest, they will gradually separate, and finally attain equilibrium with the lighter liquid above the heavier. But if two gases are placed in communication with each other, even if the heavier be below the lighter, they will rapidly intermingle MIXTURE OP GASES. 107 until the proportion of the two gtitics is the same throughout, and the greater the difference of density the more rapidly will the mixture take place. Take two different gases, of the same temperature and pressure, contained in separate vessels ; let a communica- tion be established between the vessels, and it will be found that, unless a chemical action take place, the two gases will permeate each other till they are completely mixed, and that, when equilibrium is attained, the pressure of the mix- ture will be the same as before, provided the temperature is the same. Hence, from this experimental fact, the follow- ing proposition can be deduced. 57. Mixture of Equal Volumes of Oases having Unequal Pressures.— 7/ two gasen hnvlng the same temperature he mixed together in a vessel of volume p, and if the pressures of the gases when respectively con- tained in V, at the same temperature, be p and p', the pressure of the mixture will be p +p'. Suppose the gases are separate. Take the gas whose pressure iap, and change its volume until its pressure is p', its temperatuie remaining the same. Its volume will then be, by Mariotte's law (Art. 48), —7 v. Now let the two gases be mixed without change of vol- ume, so that the volume of the mixture is , P P +P' P V then the pressure of the mixture will be p\ according to the preceding experimental fact (Art. 56). Now if the mixture be compressed till its volume is v, its temperature remain- ing constant, the pressure will become, by Mariotte's law, p ■\-p'. This result is equally true for a mixture of any number of gases. I i 108 VAPORS, OASES. 58. Mixtnre of Unequal Yolumes of Gases hay- ing Unequal Pressures.— T^^^o volumes v, v\ of dif- ferent gases, at the respective pressures p, p', are mixed together so that the volume of the mixture is V ; to find the pressure of the mi,\ ture. Change the volume of each gas to F; their pressures will be, respectively (Art. 48), V and therefore (Art. 57) the pressure of the mixture is V , v' , yP + yP 'y and if P be this pressure, we have PV = pv -\-p'v'. ' (See Besant's Hydromechanics, p. 114.) 59. Vapors, Oases. — The term vapor is applied to those gaseous bodies, such as steam, which can be liquefied at ordinary pressures and temperatures ; while the word gas generally denotes a body which, under ordinary conditions, is never found in any state but the gaseous. The laws al''eady stated of gases are equally true of vapors within certain ranges of temperature, the only diflference between themechanical qualities of vapors and gases, as distinguished from their chemical qualities, being that the former are easily condensed into liquids by lowering the temperature, while the latter can be condensed only by the application either of great pressure or extreme cold, or a combination of both. Prof. Faraday succeeded in condensing a number of different gases ; he found that carbonic acid, at the temperature of —11'', was liquefied by a pressure of 30 atmospheres, but wlien it was at the temperature i f FORMATION OF VAPOR, SATURATION. 109 P of 0", a pressure of 36 atmospheres* was required to produce conden- sation. In 1877, M. Pictet succeeded in liquefying oxygen by subjecting it to a pressure of 800 atmospheres ; at the close of the same year, M. Cailietet effected the liquefaction of nitrogen, hydrogen, and atmos- pheric air. Such experimental results point to the general conclusion that all gases are the vapors of liquids of different kinds, f 60. Formation of Vapor, Saturation. — The major- ity of liquids, when left to themselves in contact with the atmosphere, gradually pass into the state of vapor and dis- appear. This phenomenon occurs much more rapidly with some liquids than with others. Thus, a drop of ether dis- appears almost instantaneously ; alcohol also evaporates very quickly; hut water evaporates much more slowly. If water be introduced into a space containing dry air, vapor is im- mediately formed ; if the temperature be increased, or the space enlarged, the quantity of vapor will be increased ; but if the temperature be lowered, or the space diminished, some portion of the vapor will he condensed; in all cases the pressure of the air will be increased by the pressure due to the vapor thus formed. The formation of vapor is inde- pendent of the presence of air or of its density, the only effect which the air produces being a retardation of the time in which the vapor is formed. If water be introduced into a vacuum, it is instantaneously filled with vapor, but the quantity of vapor is the same as if the space had been originally filled with air. While the supply of water remains, as a source from which vapor can be produced, any given space will be always saturated with vapor, i. e., there will be as much vapor as the tempemture admits of. If the temperature be lowered, a portion of the vapor will be immediately con- densed, and become visible in the form of a liquid ; but if * An atmosphere denotes the pressure due to a column of mercury S9.9 inches in height. t Besant's Hydrostatics, p. 136. . . .^ ilO CHANGE OF VOLUME AND TEMPEHATURE. flie temperature be increased so that all the water is turned into vapoi, then for this and all higher temperatures the pressure of the vapor will change in accordance with the same law which regulates the connection between the press- ure and temperature of gases (Art. 53). The atmosphere always contains more or less aqueous vapor, and if p be the pressure of dry air, and tt of the vapor in the atmosphere at any time, the actual pressure of the atmosphere is /? + 7r. 61. Volume of Atmospheric Air without its Va- por. — Having given the pressures of a volume v of atmospheric air, and of the vapor it contains, to find the volume of the air without its vapor at the same pressure and temperature. Let P be the pressure of the atmosphere and p that of the vapor; and let v' be the required volume of the air withe. ^' its vapor, at the pressure P. Then P —p \b the pressure of the air alone when its volume is v. Hence we have (Art. 48), P : P — p :: V : v' ; ; ■ . ' - f"-p , . • 1/ "^ j-^ t • ■ t. 62. Pressure cf Gas wheu Volume and Temper- ature are Clianged. — A gas contained in a closed vessel of volume v is in contact with water, and its pressure at the temperature t is P ; it is required to determine its pressure when v is changed to v' and t to t'. Let p and p' be the pressures of the vapor at the temper- atures t and t', respectively, and P' the required pressure. Then P —p and P' — p' are the pressures of the gas alone, under the two sets of conditions stated. Hence, f RXAMPLE. Ill calling p and p' the uensities of the gas, we have, from (3) of Art. 54, P -p = kp (1 + «/), P'-p' = kp'{l-{-at'); also, from (1) of Art. 48, we have vp = v'p'. P' —p' _ v_ \±_^i •*• P -p ~ v'l + «r (1) which gives the value of P\ Cor. — If o and a' be the densities of vapor under the two conditions, we have y _ q^(l + «0. |it ~ P'p, v'a' will exceed va ; i. e,, more vapor will have been absorbed ' -^ the gas. But if Pp < P'p, then v'a' will be less than va, and the gas must therefoie, in changing its volume and temperature, have lost a portion of its vapor. (See Besant's Hydrostatics, p. 138.) BXAMPLE. Having given the pressures P and p ot a, volume v of atmospheric air, and of the vapor it contains, to find the volume of the air, without its vapor, at the same pressure P, the temperature remaining constant. Ans. Volume of air = — p^ t^- 112 PRESSURE OE VAPOR IN THE AfR. 'I 63. Formation of Dew, the Dew Voint,— Dew is the name given to those drops of wuier which are seen in the morning on the leaves of plants, and are especially noticeable in the spring and autumn. If any portion of the space occupied by the atmosphere be saturated with vapor, i e.f if the density of the vapor be as great as it can be for the temperature, then the slightest fall of temperature will produce condensation of some portion of the vapor ; but if the density of the vapor be not at its maximum for tbat temperature, no condensation will take place until the tem- perature is lowered below the point corresponding to the saturation of the space. If any body in contact witii the atmosphere be cooled down until its temperature is below that which corresponds to tlie saturation of the air around it, condensation of the vapor will take place, and the condensed vapor will be deposited in the form o^ dew upon the surface of the body. Heat radiates from the ground, and from the bodies upon it, and unless there are clouds from which the heat would be radiated back, the surfaces are cooled, and the vapor in the adjacent stratum of the atmosphere condenses and falls in small drops of water on the surface. The formation of dew on the ground depends therefore on the cooling of its surface, and this is in general greater and more quickly effected when the sky is free from clouds. This accounts for the dew with which the ground is covered after a clear night. A covering of any kind will diminish the formation of dew beneath ; for instance, but very little dew will be formed under the shade of large trees. The deic-point is the temperature at which vapor begins to be deposited in the form of dew, and it must be deter- mined by actual observation. 64. Pressure of Vapor in the Air.— Tables* have * Besant'g Hydrostatics, p. 143. MAXIMUM DENSITY OF WATER. 113 been formed and empirical formul{« constructed for deter- mining the relation between the temperature and the elastic force of vapor, at the saturating density, for certain ranges of temperature. If, therefore, the dew-point by ascertained, we can at once determine the pressure of the vapor in the air by means of these tables. For, if t' be the dew-point, and p' the corresponding pressure, then at any other tem- perature t of the air above t', we have, for the required pressure, 1 ■{- at V = 1 ■\-at' V 65. Eifect of Compression or Dilatation on the Temperature of a Gas.— It is an experimental fact that, if a quantity of air be suddenly compressed, its temperature is raised ; and that, if the compression be of small amount, the relative increase of temperature is proportional to the condensation. Thus, if the density be changed from p to p', the increase of temperature is proportional to 9' -9 - - i .. If the air be allowed to dilate, its temperature is dimin- ished according to the same law. A. stream of compressed air when issuing from a closed vessel is sensibly chilled. The reason that the compression or dilatation must be sudden, is that no heat should be allowed to escape, or to be admit- ted. If the experiment be performed in a non-conducting vessel, there is no necessity for rapidity of action. 66. Expansion of Bodies — Maximum Density of Water. — In general, all solid and liquid bodies expand under the action of heat, and contract when heat is with- drawn. The expansion of mercury is proportional to the increase of temperature, within certain limits; this is also the case with solid bodies, such as glass and steel. For 114 THERMAL CAPACITY— SPECIFIC HEAT, 'I ( water and aqueous bodies generally, the law of expansion is unknown. It is a remarkable property of water that, at a tempera- ture of about 4° 0. or 40° F., its volume is a minimum and therefore its density is a maximum;* and whether its tem- perature increases or decreases from this point, the water expands in volume. When the temperature descends to the freezing point, there is a still further expansion at the moment of congelation ; for this reason, ice floats in water. We can now see what takes place in a pond of fresh water during winter. The fall of temperature at the sur- face of the pond does not extend to the bottom, where the water seldom falls below 4° C, whatever may be the exter- nal temperature. As the temperature at the surface de- scends, the water at the surface cools, and being contracted, it becomes heavier than the water beneath, and sinks to the bottom. The water from beneatli rises and becomes cooled in its turn; and this process goes on till all the water has attained its maximum density, i.e., till its temperature is 4° C. But when all the water has attained this tempera- ture, it will remain stationary ; and any further cooling of the water at the surface will expand it, until it f'njilly con- geals. It is clear that the deeper the water is, ti. longer will be the time before the whole of the water has attained its maximum density, and therefore that ice will form much less rapidly on the surface of deep than on the surface of shallow ponds. It is from the fact that water expands in freezing, taken in connection with the low conducting power of liquids generally, that the temperature at the bottom of deep ponds remains moderate even during very severe cold, and that the lives of aquatic animals are preserved. * The results of Playfair and Joule give 3''.945 C. as the tempemture at which tb« density is a mazlmam. Phil. TransActions, 1866. .... ,. THERMAL CAPACITY— SPECIFIC HEAT, 116 67. Thermal Capacity — Unit of Heat — Specific Heat. — The thermal capacity of a body is the quantity of heat required to raise the temperature of the body one degree. The unit of heat which is generally employed is the quan- tity of heat required to raise a unit mass of water through one degree C, the temperature of the water being between 0° C. and 40° C. It is called the thermal unit Centigrade. The specific heat of a body is the thermal capacity of a unit of its mass ; and it is always to be understood that the same unit of mass is employed for the body as for the water mentioned in the definition of the unit of heat. Therefore, specific heat is independent of the unit, and is merely the ratio of the quantity of heat required to increase by I'' the temperature of the body to the quantity of heat required to increase by 1° the temperature of an equal mass of water. The quantity of heat expended in changing the tempera- ture from t to f and varies as t' — t when the mass is given, varies as the mass when t' — t \& given ; and therefore generally it varies as m {f — /), if m be the mass. Hence, the quantity of heat expended in changing the temperature of the mass m from t to t' is sm {t' — t), (1) where s is the specific heat of the substance, since it is the quantity of heat required to raise by 1" the temperature of the unit of mass, which may be shown by putting w = 1 and t' — t = 1. Let tlH denote the quantity of heat whicli produces in the unit of mass a change of temi)erature dt^ then the meas- fjft urQ of the specific heat is -^r* 116 SPECIFIC HEAT AT CONSTANT PRESSURE. I 68. Comparison of Specific Heat at a Constant Pressure with that at a Constant Volume.— In the specific heat of gases there are two cases to be considered : (1) when the pressure remains constant, the gas being allowed to expand ; (2) when the volume is constant. Let the pressure j» remain constant while the application of a small quantity of heat H increases the temperature T by T, and changes the density from p to p'. From (3) of Art. 55, by putting ka = K, we have p = KpT= Kp'{T+ t). (1) Now if the air be rapidly compressed into its original volume, its temperature will be increased (Art. 65), and we shall have the increase of temperature _ p — p' = /4[by(i)], where /t* is a constant. .*. the increase of temperature = fir, (2) and hence the whole change of temperature produced by the heat H, when the volume is constant, = r -f l^i-^ = ^T. (3) In order, therefore, to produce a change of temperature T when the volume is constant, the quantity of heat required IT is -T- , and consequently, specific heat at constant pressure __ H specific heat at constant volume ~ H ~ ' ^ ' 1 Cor. — Therefore the specific heat at a constant pressure exceeds the specific heat at a constant volume j and this EXAMPLES. 117 Jtant n the ered : being ation ire T :3)of (1) ?inal i we (2) I by (3) :ure ired (4) ure bis excess from (2) is equal to the quantity of heat t*T that is disengaged when the gas is suddenly compressed into its original volume. ScH. — The value of A is found experimentally to be con- stant for all simple gases, its value being approximately 1.408. (See Besant's Hydromechanics, p. 118.) EXAMPLES. 1. A mass m^ of a substance of specific heat «, and tem- perature /,, is mixed witli a mass Wg of a substance of spe- cific heat Sg and temperature t^, the mixture being merely mechanical, so that no heat is generated or absorbed by any action between the substances, and all gain or loss of heat from external sources is prevented. Find the resulting temperature t of the mixture. Suppose the former body to be the warmer ; then it cools down from t^ to ty while the colder rises from /g to t. Therefore we shall have m-^^Si {t^ — /) = the units of heat lost by the former body. and mgSjj {t — t^) = the units of heat gained by the latter body. and since the quantity of heat lost by the warmer body is equal to that gained by the cooler, these two expressions are equal ; therefore wiiSi (/j — = ^2*2 (^ — ^2) ; One of the methods of finding the specific heat of a sub- stance is by immersing it in a given weight of water, and observing the temperature attained by the two substances. 1' 118 SUDDEN COMPRESSION OF A MASS OF AIR, 3. A mass M of a substance of specific heafc S and tem- perature Ty is immersed in a vessel of water, m' and m being the musses of the vessel and of the water in it, and t' their common temperature and s' the specific heat of the vessel. Find the temperature t of the whole after immer- sion. . , _ MST + ml' + 7wV/' Ans. t - j^^ _^ ^ _^-^^, . 3. A glass vessel weighing 1 lb. contains 5 oz. of water, both at 20°, and 2 oz. of iron at 100° is immei*sed. What is the temperature of the whole, taking .2 as the specific heat of glass and .13 of iron ? Ans. 22°^. The following are approximate values of the specific heats of a few v'iubstances : Water, 1 Thermometer-glass, .... 0.198 Iron, 0.114 Zinc, 0.1 Mercury, 0.03 Silver, 0.06 Brass, ........ 0.09 (Besant's Hydrostatics, p. 147.) 69. Sudden Compression of a Mass of Air.— .i inass of air being suddenly * compressed or dilated, it is required to find the new pressure and temperature. Let p, p, T be the pressure, density, and absolute tem- perature at f ny stage of the process ; p\ p', T the new pressure, density, and temperature ; an^ let dTht the change of temperature due to the change dp in p. Then we have * If the comprcBsion takes place in a non-condncting vessel, so that no heat is lost or gained, the compressioa need not be rapid. \\ SUDDEN COMPRESSION OF A MASS OF AIR, 119 From (1) of Art. 68, we have p = KpT; = AT + K^T [from (1)]. Dividing (3) by (2), we have (2) (3) i ^^ = 1 + f* = ^ [from (3) of Art. 68], pdp p p p V dp __ kdp 'p ~' P ' or, - = — (4) Integrating between the limits jw' and p, p' and p, we have P' _ (plY. J^\pr (5) which determines the pressure. Also, p' = Kp'T, which, divided by (2), gives p' __ p'T' $ "^ pT From (5) and (6), we have r _ /p'\x-i T "ypr ... r=Tn (6) \-l (7) which determines the temperature. (See Besant's Hydro- mechanics, p. 116.) laO U EIGHT OF THE HOMOOEyEOUS ATMOSPHERE, 70. Mass of the Earth's Atmosphere.— By moans of the barometer, some idea may be formed of the mass of air and vapor surrounding the earth, since the weight of the whole atmosphere is equal to tliat of a stratum of mercury about 29.9 inches thick covering the globe. Suppose the earth to be a sphere of radius r, and that h is the height of the barometric column at all points of its surface. 1'hen the mass of the atmosphere is approximately equivalent to the mass ^"narVi, of mercury, where o is the density of the mercury. Let p be the mean density of the earth ; then, the mass of the atmosphere : the mass of the earth = ^TTor^h : |7rp?** = 3ff/i : pr. Taking a = 13.568 (Art. 47), and p :_i 5.5,* and sup- posing the height of the barometric column h to be 30 inches, which is probably near the average height at sea- level,! it will be found that the above ratio of the mass of the atmosphere to that of the earth is about TirWirff' 71. The Height of the Homogeneous Atmosphere. — If the atmosphere were of the same density throughout as at the surface of the earth, its height I would be approx- imately obtained from the following equation, ah ^ plf (1) where o and p are the densities of mercury and air respect- ively, and h is the height of the barometric column. From Art. 70, and Art. 33, Sch., we have or = 13.568 x7G8p = 10420.324p, * There \» some doubt about the accuracy of thi? vahic ; the value deduced by the Astronomer Royal at the Harton Colliery in 1864 is 66. Phil. Trans., 1856. t See Bncy. Brit., Vol HI., p. 28. I'. \ LIMIT TO THE U EIGHT OF THE ATMOSPHERE, 121 and taking h = 30 inches, we have, by solving (1) for /, I l = h- = 26050 feet, 9 which is a little less than 5 miles. Ti» Necessary Limit to tlie Heiglit of the At- mospliere. — Since the attraction of the earth diminishes at a distance from its surface (Anal. Mechs., Art. 133a), it is clear that the atmosphere is very far from being of uni- form density throughout, and therefore the result in Art 71 is very far from the truth. A limit can be found, however, to the height of the atmosphere from the consideration that, beyond a certain distance from the earth's centre, its attrac- tion will be unable to retain the particles of air in the cir- cular paths which they describe about the earth, since the centrifugal force must exceed the force of gravity. Let o) be the earth's angular velocity, and r its radius. Then the centrifugal force of a particle m of air on the earth's surface is mwV, and this is equal to ^^ [Anal. Mechs., Art. 199, (3)] ; therefore, at a height z above the surface, the centrifugal force mw* (r -f z) _ mg r + z ~ 289 ~r The earth's attraction at the same height (Anal. Mechs., Art. 133«) __ mgi^ -(T+^p' and, in order that the particle may be retained in its path, these two forces must equal each other. mg r ■}- z _ mgr^ 122 DECREASE OF DENSITY OF THE ATMOSPHERE, 'mt' or, r -\- z 289r ra z = r(V389-l); = 5.6r + = 33000 mile^ (approximately). Rem. — The actual height of the atmosphere, however, is possibly much lower than this, for its temperature has been found, by experiments made in balloons, to diminish with great rapidity during an ascent; it is therefore very likely that, at a height less than 5r, the air may be liquefied by extreme cold, and in that case its external surface would be of the same kind as the surfaces of known inelastic fluids. (Besant's Hydromechanics, p. 120.) 73. Decrease of Density of the Atmosphere.— (1) WTien the force of gravity is constant. Take a vertical column of the atmosphere, and let it be divided into an indefinite number of horizontal strata of equal thickness, so that the density of tlie air may be uni- form throughout the same stratum. Let the weight of the whole column from the top of the atmosphere to the earth = «, that of the whole column above the lowest stratum = h, that of the column above the second = c, and so on. Then h, c, «?, etc., are the forces respectively which compress the first, second, third, etc. strata, which, as they are of equal thickness, are as their weights, a — hy b — c, c — d, etc. Hence we have a — h '. h — c '.'. h : c\ ,\ a X h '.: h '. c. ' •' In the same way, it may be shown that h : c \\ c \ dy and so on, Hence, d, Cy d, etc., and therefore the densities I ^ DECREASE OF DENSITY OF THE ATMOSPHERE. 123 of the successive strata, form a series of terms in geometric progression, which is decreasing since a is greater than J, and therefore h greater than c, and so on ; and as the strata all have the same thickness, the heights of the several strata ahove the earth's surface increase in arithmetic progression. Hence, ; - >v ,; , // a sei'ies of heights be taken in arithmetic pro- gression, ivhen the force of gravity is constant, the densities of the air decrease in geometric progression. ScH. — By barometric observations at different altitudes, it is found that at tho height of 3|^ miles above the earth's surface, the air is about one-half as dense as it is at the surface. Forming therefore an arithmetic series, with 3^ for the common difference, to denote the heights, and a geometric series with | for the common ratio, to denote densities, we have . • Heights, 3|, 7, 10|, 14, l^, 31, U\, 28, 31^, 35, etc. Densities, j, \, \, t^, ^, ^, ^^, ^^, -g^, jisWjetc. That is, according to this law, at the height of 35 miles the air is less than a thousandth part as dense as it is at the surface of the earth. ^ (2) Jfhejt the force of gravity varies inversely aj the square of the distance from the earth's centre. Let r be the radius of the earth, p' the density at the sur- face of the earth, p the density at a height «, and h the height of a homogeneous atmosphere. Then, since the density varies as the compressing force, and this varies as the weight, we have p' '. dp '.', hp'g '. gj-~-^^{-pdz), . . , , yihexQ g and g^ {r + zf are the measures of the earth's attract t lli 124 HEIGHTS DETERMINED BY THE BAROMETER. tion at the surface and at a height «, the negative sign being taken because the density is a decreasing function of the height z. dff _ r^ dz p ~ h {r -\- z)^ Integrating, observing that when z = 0, p = p', we iive ^''^^==h(7Tz--rh P = + r* /I _ J_\ » A vf ~ r+«/ I which shows that, if r + 2 increases in harmonic progres- sion, will decrease in arithmetic progression, and r -{■ z therefore p will decrease in geometric progression. Hence, If a series of heights be taken in harmonic progres- sion, when the force of gravity is regarded as variable, the densities of the air decrease in geometric progres- sion. (See Bland's Hydrostatics, p. 358.) 74. Heights Determined by the Barometer.— A very important use of the barometer is to find the difference of level of two places situated at unequal distances above the surface of the earth. Since the height of the column of mercury in the barometer depends on the pressure of the atmosphere (Art. 43), and as the pressure of the atmosphere at any point depends upon the height of the column of air extending from that point to the top of the atmosphere, it iollows that this pressure will decrease as we ascend above the earth's surface, and therefore that the height of the column of mercury will diminish. That is, the mercury in the barometer will fall when the instrument is carried from I HEIGHTS DETERMINED DY THE BAROMETER. 125 2ing the we - the foot to the top of a mountain, and will rise again when it is returned to its former position. (1) When the force of gravity is regarded as con- stant. Consider a vertical column of the atmosphere at rest under the action of gravity. Let z be taken vertical and positive upwards ; and at a height z, let p be the pressure and p the density. The pressure jo, at any height z, is meas- ured by the weight of the column of air extending from that height to the top of the atmosphere; and the element- ary pressure dp will be measured by the weight of the col- umn having the same base and the elementary height dz. Therefore, if -4 be the area of the section of the column, we have Adp = — Agp dz, or, dp = —gpdz. (1) the negative sign being taken because the pressure p is a decreasing function of the height z. If ^ be the temperature, we have from (3) of Art. £4, p = kp{i -\- at). (2) Dividing (1) by (2), we have gdz p \ ^ at (3) If the heights above the earth's surface are small, the force of gravity g may be regarded as constant ; and sup- posing t constant, we have, by integrating (3), where ^' is the pressure at the height z'. (4) 126 HEIGHTS DETERMINED BY THE BAROMETER. Let hf h' be the observed barometric heights at the two stations, whose altitudes are z and z' ; let a be the density of mercury at a temperature zero, and t, t', the temperatures at the two stations. Then we have, from (2) of Art. 53, p = gah (1 — 0r), and p' = gah' (1 - Br'), which in (4) gives (5) where t may be taken approximately equal to ^ (t + r') ; from this equation the difference of the heights of the two stations can be calculated. (2) When the force of gravity is regarded as va- rifd)le. If the heights above the earth's surface be considerable, it is necessary to take account of the variation of gravity at diflEerent distances from the earth's centre. Calling g the measure of the earth's attraction at the level of the sea, and r the radius of the earth, then we have, for the measure of the attraction at a height «, <; ^{r + zf which, being substituted in (1) for//, gives r2 dv--g p dz. {r + zf Dividing (7) by (2), we have , dp __ 1 gi^dz p "" "" W^i (r 4- %f (6) (7) (8) ■ i two tyof ures («) 'HEIGHTS i>ET^kMINED BY THE BAROMETER, 127 It milst be observed that /; is the sum of the pressures due to the air itself, and to the aqueous vapor which is mixed with it ; i. e., the quantity kp in (2) is the sum of the two, Jcp, k'p', where p.und p' are the densities of the air and , the aqueous vapor, respectively. - : . Considering t constant as before, and equal to the mean of the temperatures at the two stations, and integrating (8), we have *'"«/, =-(i-T-«0 ('• + *)('• +7)- <"> As before, let //, h', and r, t', be the observed barometric heights and temperatures, and a the density of mercury at a temperature zero ; then from (2) of Art. j2, by sut)stitut- ing for g its vilue from (6), w6 have p Vr + zf 1 — St. h . ..:: ^, ;■ , . . .^. ./ Substituting (10) in (9), and solving for z — z\ we have z-z'= ■' '^• k{\ + at)(r-\-z){r-\-z')L A' ^ , r+z. \-er\ .• (11) Since is very small (Art. 53), we have * 1 - er log Or = log [1 - B {r[- r)] = _0(r'_r). '^>. v V '..V (Calculus, Art. 61.) 128 HEIGHTS DETERMINED BY THE BAROMETER. Substituting this in (11), and reducing Naperian to com- mon logarithms by multiplying by w, the modulus of the common system, we have z — z=. h{l-{-ttt){r-\-z){r-{-z mgr^ ->[iog4 ^+2log,o— , me{T' — t) (12) from which the value of z can be determined when z' is known. Cor. 1. — If the lower station be nearly at the level of the sea, z' = 0, and (12) becomes (13) Cor. 2. — In the above investigation no account has been taken of the variation of gravity at different parts of tlie earth's surface. From a comparison of the results obtained by causing pendulums to oscillate in different latitudes, if ^f be the measure of gravity at a place of latitude A, and g' at a place of latitude a', it has been found (Poisson, Art. 628) that ^ _ 1 — .002588 cos 2 k , g' ~ I — .002588 cos 2A,' ' h hi— .002588 cos 2A' (14) therefore, ^^ _ ^^, ^ _ ^^^^^^ ^^^ ^^ If A' be the latitude of Paris, the value of the quantity h mg 7 (1 - .002588 cos W) (15) is nearly 18336 French metres,* or about 60158.56 English * A French metre \% 89.37079 iiicbeB, ;om- the fl2 / is '] HEtOBTS DETBRMIXED BY THE BAROMETER. 129 feet; representing this numerical quantity by c and substi- tuting it in (14), we get k c 1 — .002588 cos 2A' which in (13) gives z = (n-«o(] l-.002588co^r^'V7+^^"^'»l^ + r) — rnO (r '-T)], (16) from which the value of z may be determined by a series of approximations; i. e., an approximate value must be first z obtained by neglecting - ; then this approximate value * must be substituted for 2; in -, and a more accurate value r will be obtained, and the same process may be repeated, if necessary.* - ScH. 1. — When - is very small, it may be neglected in (16). It has been found in practice, however, that in this case the results are more accurate by employing 18,393 metres as the value of c. (Duhamel, p. 259.) In order that the heights as determined by the barometer may be very exact in practice, certain corrections are neces- sary. For instance, the value of k is modified by the fact that the density of aqueous vapor at a given temperature and pressure is less than the density of dry air under the same circumstances ; and the proportion of aqueous vapor to dry air will generally be different at the two stations. * A formala for tbis is given in Bncy. Brit., Vol. III., p. 8(i6, involvlDg a consid- eration of densities of vapor. 130 SPECIFIC GRAVITIES. ScH. 2.— Formula (16) has been obtained on the supposi- tion that the temperature of the air remains constant in passing from the lower to the higiier station ; if, however, the diflference between the heights be very grea\ a consid- erable error may be thus introduced, and for- sulae have therefore been constructed in which account is taken, on various hypotheses, of the variation of atmospheric temper- ature. A foi.aula of this kind is given in Lindeman's Barometric Tables, constracted on the supposition that the temperature diminishes in harmonic progression throujj;h a series of heights increasing in arithmetic progression. Also, " i ume, and the temperature changed to /', find the pressure P' on the piston. , _ ,,, ,.^ 1 4- tW * .INS. I =z J tl — • •» , 1 + «* 18. If a cubic foot of gas, whose tem])eraturt is 100° and elastic force 29^ inches, be cooled down to 40 , and com- pressed by a force equivalent to lOi^ inches, find ils volume. Ans. 4334.7 cubic nches. 19. If 20 cubic inches of air, whose temperature is 50° and elastic force 28.8 inches, be expanded to 25 inches by the application of heat, and if the elastic force become 31 inches, find the temperature. Ans. 234. 27"". 20. Let 100 cubic inches of air have a temperature 32° and a pressure 29.922 inches; if the temperature become 60°, and the pressure 30 inches, find the volume. Ans. 105.42 cubic inches. 21. A cubic foot of air at a temperature of 100°, and under a pressure of 29|^ inches of mercury, is cooled down to 40° and compressed by an additional 10^ inches of mer- cury. Find the volume. Ans. 1137.80 cubic inches. 22. If h and h' be the heights of the surface of the mer- cury in the tube of a barometer above the surface of mercury in the cistern at two different times, compare the densities of the air at those times, the temperature being supposed unaltered. Atis. h : h'. 23. A conical wine-glass is immersed, mouth downwards, in water. How far must it be depressed in order that the water within the glass may rise half-way up it ? Ans. Ih, where // is the height of the water barometer. 24. A cubic foot of air having a pressure of 15 lbs. on a square inch is mixed with a cubic inch of compressed air, having a pressure of 60 lbs. on a square inch. Find the pressure of the mixture when its volume is 1729 cubic inches. Ans. IS-i^l^ lbs. KXAMPLKS. 135 25. Two volumes, V luul F, of different gases, at press- ures p p\ and temperature t, are mixed together ; the vohime of the mixture is l\ and its temperature t'. Deter- pV + p'V 1 -\- at' mine the pressure. A UH. U I + at 36. Three gallons of water at 45° are mixed with six gal- lons at 90^ What is the temperature of the mixture ? An». 75° 27 An ounce of iron at 120°, and 2 oz. of zinc at 90°, are thrown into 6 oz. of water at 10°, contained in a glass vessel weighing 10 oz. What is the final temperature, taking .1 and .12 as the specific heats of zinc and iron ? Ans. 13 ffi- PART II. HYDROKINETICS. M CHAPTER I. MOTION OF LIQUIDS.— EFFLUX. — RESISTANCE AND WORK OF LIQUIDS. 75. Velocity of a Liquid in Pipes.—// a liquid run through any pipe of variable diameter, wJiicli is kept continually full, and the velocity is the same in every part of a transverse section, the velocities in the different transverse sections vary inversely as the areas of the sections. For as the tube is kept full, and the liquid is incom- pressible (Art. 3), the same quantity of liquid which runs through one section will, in the same time, run through the next section, and so on through any other. Hence if h, h' be the areas of any two sections, and v, v' the veloc- ities of the particles at those sections, we have, since the quantity of liquid which flows through any section in a unit of tin:e is the product of the area of the section by the velocity, kv = k'v' ', V : V iC \ IC% (1) Cor. — Hence, as the section of a mass of liquid decreases, its velocity increases in the same proportion. For instance, the velocity of a stream or river is greater at places where its width is diminished. This demonstration is also applicable to different sections of a liquid issuing through the orifice of a vessel, whether tlie section be taken within 'U VELOCITY OF EFFLUX. 137 or without the vessel, provided there be no vacuity in the stream between the sections. ScH. — It is supposed in this proposition that the changes in the diameters of the sections are gradual^ and nowhere abrupt ; if there are any angles in the pipe, they will proi^uce eddies in the motion of the liquid, and the propo- sition v.. li not hold true. 76. Velocity of Efflux.—// a small apeHure he made in a vessel containing liquid, the velocity with which the liquid issues from the vessel is the same as if it had fallen from^ the level of the surface to the level of the aperture.* Let EF represent a very small orifice in the bottom of the vessel ABCD, which is filled with a liquid to the level AB; and suppose the vessel to he kei)t full by supplying it from above, while the liquid is running out through the orifice EF. Let t be the velocity of efflux, w the weight of the liquid which issues with that velocity per second, and h the height of the surface above the orifice, called the l:(>nd\ of the liquid. Then the work which w can perform while descending through the distance A, from the surface to the orifice = why and the kinetic energy stored up in to as it issues through the orifice w = -- t'2 (Anal. Mechs., Art. 217). If we suppose there is no loss of energy during the passage through the orifice. ♦ TIiIh is known as Tonicelli'e Tlieoiein. t Tilt term fuuU in HydruineclianiCB is nieasuriMl, rolafively to any i)oinf, by the depth of that point holow the purfacc of the liquid. Since the liquid in F\g. 34 (l(>«ay there arc h feet of head above the orifiQo. • i 138 VELOCITY OF EFFLUX. we may equate these two quantities of work, and shall have from which we find -«7 = '^*' V* ^gh; V = V2gh; (1) (2) tluit is, the velocity of efflux is the same as that of a body which has fallen freely through the height h. From (2) we have A = — , in which the height //, corresponding to the velocity v, is called the head due to the velocHy, or simply the head. The corresponding velocity is called the velocity due to the head. Cor. 1.— If the orifice be made in the vertical face of tlie vessel, and a tube be inserted so as to direct the current obliquely, horizontally, or vertically uptvard, the velocity of efflux will be the same, since the pressure of fluids at the same depth is the same in every direction (Art. 7), and each particle of liquid having the same velocity will follow the same path ; a parabola whose directrix, whatever be the angle of elevaticn, is fixed, and lies in the surface of the liquid (Anal. Mechs., Arts. 151 and 153). If the liquid issue obliquely, its equation is given in (3) of Art, 151, Anal. Mechs. If it issue horizontally, « = 0, and this equation becomes jr = — y =■ 4hy. if Cor. 3. — If h^ be the depth of a second orifice below the surface, and y, the velocity, we have therefore, from (2) and (3), we have y : V, :: a/A : y/h^\ (3) VELOCITY OF EFFLUX. 139 that is, the velocities of efliur' are as the square roots of the depths. Cor. 3. — The quantity of liquid run out in any time is equal to a cylinder, or prism, whose base is the area of the orificR, and whose altitude is the space described in that time by the velocity acquired in falling through the height of the liquid. Cor. 4. — If any pressure be exerted on the surface of the liquid, the velocity of efflux will be increased. Let h be the depth of the orifice below the surface of the liquid, ^1 the height of the column of liquid which would exert the same pressure as that which is applied at the surface; then the velocity of efflux will be due to the vertical height h •\- h^', hence we have from (2) V = ^/%g{Ji + h^). (4) If /aj be taken equal to the height of a column of water e(|ual to the pressure of the atmosphere (=34 feet), (4) becomes •_ ,• v= Va^r (A -r34). . (5) ivhich is the velocity of efflux luhen a liquid is pro- jected into a vacuum, the orifice being at a depth //, below the surface of the liquid. If A^ be the area of the orifice, then the quantity of liquid Q which flows through the orifice in the unit of time is Q = Tcv = h y/'igli. (6) Cor. 5. — If a parabola, with a parameter = ^g, be described with its a.^is vertical, and vertex in the upper surface of the liquid, the velocity of efflux through any .small orifices in the side, would be represented by the cor- responding ordinate?. ;. 140 THE HORIZONTAL RANGJU. ScH. — The correctness of this theorem can also be shown by the following experiment. If in the vessel (Fig. 34) an orifice K or R be made, directed vertically upwards, the velocity of the jet K or R is such as to carry the particles of liquid up nearly to the same level as the surface of the liquid in the vessel. Practically the resistance of the air and friction in the conducting tube destroy a portion of this velocity. EXAMPLES. 1. With what velocity will water issue from a small orifice 16-^j ft. below the surface of the liquid ? Ans. 32|ft. 2. A vessel has in it a hole an inch square ; water is kept in the b.isin at a constant level of 9 ft. above the hole; what is the outflow in one hour? Ans. 600 cu. ft. 3. What is the discharge per second through an orifice of 10 square inches, 5 ft. below the surface of the liquid ? Ans. 2152 cu. ins. 77. The Horizontal Range of a Liquid Issuing tTirough a very Small Orifice in the Vertical Side of a Vessel.— Let ABCD be a vessel filled with a liquid, having its side BC vertical, M a small orifice in the side of the vessel, MH the parabola described by the liquid, and CH the horizontal range. On BC describe the semicircle BFC, and through M draw MN perpendicular to BC. If the liquid issue horizon- tally from the orifice M, the equation of its path is (Art. 76, Cor. 1), Fig. 35 X 2 — 4Ay, (1) in which h = BM, the height of the surface above the 1) . TIME OF DISCHARGE. 141 orifice ; then the range CH will be determined by making y = MC. Hence we have from (1) a; = 3 y/hy = 2 VBM x MC = 2MN; (2) thut is, the horizontal range of a liquid issuing hori- zontally through a very small orifice in the side of ft vessel is equal to twice the ordinate at the orifice, in a semicircle whose diameter is the vertical distance from the surface of the liquid to the horizontal plane. Cor. — Wh? .1 the orifice is made at the centre of the side BC, the horizontal range is a maximum, and equal to the height of the liquid above CH; at equal distances above and below the centre, the range will be the same. 78. Time of Discharge from a Cylindrical Vessel when the Height is Constant.— When a cylindrical vessel is kept constantly full, it is required to deter- mine the time in which a quantity of liquid equal in volume to the cylinder will flow through a small orifice in its haae. Let h be the height of the surface, K the area of the base of the vessel, and k of the orifice, V the velocity of descent of the surface of the liquid, and v the velocity of efflux at the orifice, and / the time necessary to discharge a volume of li(iuid equal to that of the cylinder, which remains con- stantly full. Tiien the quantity of liquid which flows through the orifice in the unit of time is k V^f/h ; and since the velocity of the surface is V, the quantity of liquid which passes through the orifice in the unit of time must equal VK. Hence we have _ VK=zkV2gh', { 14"/J TIME OF EMPTYING ANY VESSEL. and as the vessel is kept constantly full, we have (a) Q where Q denotes the whole quantity of liquid in the vessel. Cor. — If the liquid be kept at a height // in a second vessel, containing a quantity Q'y which flows through an orifice k\ in the time t, we have from (1) ^ = — T-=; (3) hW^gh' ^ ' and from (1) and (3) we have Q'.Q'-.'.k Vfi : h' Vh'. Hence, the quantities discharged iri the same time, frovi orifices of different sizes, and at different depths, are as the areas of those orifices and the square roots of their depths jointly. 79. The Time of Emptying any Vessel through a 8 mall Orifice in the Bottom.— Let EH be the upper suriace of the liquid at the time /, x and y the distances OD and DH, h the depth OC of the liquid when the vessel is full, h the area of the orifice, and K the area of the upper surface of the liquid at the time /, which, when the figure of the vessel is known, will be given in terms of X and y. Then the quantity of liquid which flows through the orifice in an element of time is Tc '^'^gx dt ; and since in the same time the surface EH descends a distance dx, the quan- Fig. 36 TIME OF EMPTYIXO ANY VESSEL. 143 tity of liquid which flows through the orifice in this time must equal Kdx. Hence we have k y/'^gx lit = — Kdx, the negative sign being taken, because x decreases an t increases, Cor. 1. — If the vessel be a surface of revolution round a vertical axis, K = ny^, which in (1) gives t = -"/ * yMx k V^gx (2) Cor. 2. — To determine the time of emptying a right cylinder or prism. Here K is constant, and (i) becomes kV2g 2K , ._ ._. /dx 'UK 1 . ^ Vx k V2<7 (3) remembering that when >< = 0, x = h. When x =z Q, we have for the time of emptying the whole cylinder, t = J-^^Vh = -^=, (4) k V2g k V2gh where Q denotes the quantity of liquid in the vessel. By comparing this result with that in (1) of Art. 78, it appears that the time nceessarij for the entire discharge of the liquid irhert the vessel cnijMes itself is twice as great as tJint which is required to discharge the same quantitij ah en the vessel is kept constantly full. Cob. 3. — If a cylinder of given altitude empty itself ia I I ! j ■ ) '1 1 I i 1 L i 1 [ ^ 144 THE TIME OF EMPTYTAQ A CYLINDEIi. n seconds, tlirough ii given orifice, the radius r of the cylinder from (4) ii 7T V^h and if tlie radius is given, its height 7i is (5) (6) 80. The Time of Emi»t.vhig a Cylinder into a Vacuum. — 2b determine the time in which a cylin- drical vessel will einptij itself, through an orijice in the bottom, into a vacumn, when its upper surface is exposed to the pressure of the atmosphere. Let h be the l^^eight of the vessel, h' the lieight of a column of liquid which is ecjual to the weight of the atmosphere ; and x the depth of the orifice below the upper surface of the liquid. Then from (4) of Art. 76, the velocity of discharge is V^/y {x + h'), which in (1) of Art. 79 gives _ K /• ilx 2Ji Jc ^/'Ig {x + 7i')2 + G 3A' hW'^g Bince when x = A, t — 0. And making x = 0, (1) becomes which is the time of emptying the vessel. (1) (2) ix^ama CYLINDSICAL VESSEL WITH TWO SMALL ORIFICES. 145 81. The Time of Emptying a Paraboloid Let the vessel he a paraboloid of revolution round the vertical axis, h its height, and 2/> its parameter. Then if X is 'the depth of the orifice in the bottom below the upper surface of the liquid, we have y^ = 2joa:, which, in (2) of Art. 79, gives . __ 2j[>Tr I* xdx __ 4jt)7r ^/x -x% 3i- a/2^ x^ -\- G (1) since when x =z It, t ■= 0. Making a; = 0, and i)utting r = the radius of the base, (1) becomes 2 rVAl ^gj t = /r— » 3 k V2~g which is the time of emptying the vessel. 82. Cylindrical Vessel with Two Small Orifices.— A cylindrical vessel of given dimensions, is filled with a liquid ; there are two given and equal small orifices, one at the bottom, the other bisecting the altitude ; to find the Ume of emptying the upper half, suppos- ing both orifices to be opened at the same instant. Let 2« = the altitude of the vessel, x = the altitude of the surface of the liquid from the upper orifice at the time /, and r = the radius of the base. Then the quantities of liJ. Cor. 1. — As the ratio .^ of the sections decreases the A velocity decreases, becoming a minimum and = V^'iffh, when the cross-section k of the orifice is very small com- pared with that of A', which agrees with {'i) of Art. 76, as it clearly should. I' Cor. 2. — As the ratio -r^ increases tlie velocity increases, A and it approaches nearer and nearer to infinity, the smaller the difference between the two cross-sections becomes. If k = JC, (4) becomes V _ V'^gh _ co; from which we infer that, if a cylindrical vessel is without a bottom, a liquid must flow in and out with an infinitely great velocity, or else a section of the liquid flowing out of the vessel can never be equal to a section of the vessel. If a cylindrical tube be vertical, and filled with a liquid, the portion of the li(juid at the lower extremity, being urged by the pressure of all above it, will necessarily have a greater velocity than those portions which are higher, and therefore (Art. 75) a section of the liquid issuing from the vessel must be less than a section of the tube, i.e., the stream of liquid will not fill the orifice of exit* (3) ncity EXAMPLE. If water flows from a vessel, whose cross-section is 60 square inches, through a circular orifice in the bottom 5 inches in diameter, under a head of water of 6 feet, find its velocity. A?is. 20.79 ft. 85. Reetaiigiilar Orifice in the Side of a Vessel. — 1\) deterniine the quantity of liqidd irhich will * Formula (4) vt&s firt>t f;iven by Beruooilli, aod wae afterwards much dispatecl (Weisbach'a Mechanics, p. 804). Jv': V ' r ! 1 r 1 1 ■, * 150 RECTANGULAR ORIFICE IX A VESSEL, flow from a rectangular orifice in the side of a vessel which is kept constantly full. (1) mien one side of the orifice coincides with the surface of the liquid. Let h be the heiglit and b the breadth of the rectangular ori- fice ALMD, til rough which the efflux takes place ; let HK be a horizontal strip at the dis- tance X below AD, and of infin- itesimal thickness d.r, so that the velocity of the liquid in every i>art of the strip is the same. Then the velocity of efflux through this strip is V'ip" [Art. 70, (:i)], and the finantity discharged in a unit of time is hdx y/"i(ix\ hence, calling Q the whole quantity discharged in a unit of time, we have Q = fb(lx\/)l^'^ (1) and integrating between x — and x = //, we have Q = ibV'^(M (2) If we denote l)y v the tucan velocity, i.e., the velocity which would have to exist at every point of the orifice, in order that the same ((uantity of liquid would flow throngh the orifice with a uniform velocity as now flows throngh with the variable velocii ', we have which in (2) gives Q = bliv, (3) Hence Nie niodii relocitif of a, liquid floiriu^ onf through a rectangular orifice in the side of a vessel TRIANGULAR ORIFICE IN THE SIDE OF A VESSEL. 151 vessel \th the (1) (2) is I the velocity at the lower edge of the orifice ; and the qaautity of liquid flowing out through this orifice in any given time is f the quantity that would flow through an orifice of equal area placed horizontally at the whole depth, in the same timcj the vessel being kept constantly full. (2) Vrlwn the upper surface of the rectangular ori- fice is hrloir the surface of the liffuid. Let SR be tlie upper etlga of tlic oritice at the depth hy below the surface AD. Then, integrating (1) between tlie limits X ^ hi and x = h, we have If the mean velocity of efilux is r, we iiave Q = b{h-h,)v, which in (4) gives , S ,3 {^) (5) 8«. Triangular Orifice in tlie Side of a Vessel.— (I) irhrn the verfr.v of the triangle is in the surftre of the liquid. Let h be the heigiit EF, and b the breadth IIF of the triangular orifice EHF, through wiiich the efflux takes place; let LM be a horizontal strip at the distance x below AD, and of infinitesimal thickness dx, so that the velocity of tlie liquid in every part of the strip B is tlie same. ' ^'a- *° Then LM = . .r, and calling Q tlie (pumtity of licpud disclmrged in a unit of time, we have A 1 < E D m M y 1 ; I I IP 153 TRIANGULAR ORIFICE IN THE SIDE OF A VESSEL. = \hy/^h\ (1) If the mean velocity is v, we have which ill (1) gives v = f a/^^tA. (2) (2) Wlicn the base of the triangle is in the surface of the liquid. Let, KEH be the triangular orifice, KE = h, and KH = h. Then the quantity discharged through KEH will equal the discharge through the rectangle KHFE, minus that through the triangle EUF ; therefore subtracting (1) of this Art. from (2) of Art. 85, we have and = ^gb/iVigh, (3) Cor. 1. — If the orifice be a trapezoid ABCD, A F K E B whose upper base AB = ij lies in the sur- face of the li(|uid, whose lower base CD = ig, and whose altitude is DF — h, the discharge may be found by combining the discharge through the rectangle ECDF with those F«g- 4i through the two triangles ADF and BCE. Hence, combining (3) with (2) of Art. 85, we have ^T^irCJ*! ■V'^b;)hy/^h. (5) iL. "iEL. TRt ANGULAR ORIFICE IN THE SIDE OF A VESSEL. 153 (1) urfacc Id KH H will minus "g 0) (5) Cor. 2.— If the orifice be a triangle DCH (Fig. 41), whose base DC = h^ is situated at a depth KL = //^ be- low the surface of the liquid, and whose vertex H is at a depth li below the surface, the discharge is equal to that through the triangle AHB, minus that through the trap- ezoid ABCD. Hence, from (3) and (5), we have q = ^hh\^'Zgh - T?j (-iA + 3i,) h^ y/Hyh^ Since AB : DC :: UK : HL, we have b '. h^ '.'. h '. h — h^\ which in (0) gives V" k-h, )' («) Q = 15 (7) Cob. 3. — If the orifice be a trimgle ABC, whose vertex A is above its base, snd at a depth 7*1 below the surface of the liquid, whose base CB = ^i is at a depth h below the surface, the discharge is equal to that through the rectangle ACBK, minus that H through the triangle ABK. Hence, from (7) and (4) of Art. 85, we have Fig. 42 • _ 2V2g b^ /3A* - 5^^/*^ + ^hA /gv ~~ 15 V h — hi J 154 ORtFICE IN THE STDK OF A VKS8KL. Otherwise thus: Let ODC be a vertical orifice, formed by a plane curve, whose vertex is 0, at tlio depth AO below the surface of tiie licpiid. Ixjt AB = h, AO = /;^ / ' \ D 3 C Fig. 43 Q = J 'iyV'Z(j{l,^ -{- j)dx. (9) (1) When the orifice is a rectttngle. Here y is constant, wliich put = \b, and integrating (9) between the limits a; = and x ■= h — h ^, we have for the discharge through the whole orifice ODC, Q^^bV^ij{h^~h,^, which is the same as (4) of Art. 85. (10) Cor. 4. — If the upper side coincides with the surface of the liquid, h^ r~ 0, and (10) becomes ' Q ■= IbhVWh which agrees with (3) of Art. 85. (3) Wlien the oHflce is a triangle whose vertex is doirnirards and the base horizon tnl. Let a :b be the ratio of the altitude to the base ; then which in (9), and integrating between the limits x = and X — // — A^, gives ORIFTCE IN THE SIDE OF A VESSEL. 155 wliieh agrees with (7). " Coii. 5. — If the base coincides with the surface of the Ii(iuid, h^ =0, and (11) becomes t wliich agrees with (3). (3) Wlifii the orifice is a triangle whose vertex is upwards and base horizontal . Here 2y = ;^x, a which in (9), between tlie same limits, .r -- and x = 7/ — 7i,, gives which agrees with (8). (12) Cor. G.— If the \ertex coincides witli tlie surface of the liquid, hi =0, and (12) becomes Q = ^bh\/2gh, which agrees with (1). Cor. 7. — From Cors. a and 6 we see that the f,uantities discharged in the same time tlirougli two equal triangular orifices in the side of a vessel kept constantly full, the one having its base and the other its vertex upwards in tho sur- fa^^ of the liquid, are in the ratio of 2 : 3. .«=* 156 EXAMPLE. H fh 8V. The Time of Emptying ai?^^ Vessel tlirougli a Vertical Orifice.— Let A be the , rl'ace ^ of the liquid in the vessel wiieii tlie oritiee OCD is opened, and H the surface at (lie end of the time / ; let AH = ^, AO = //', OE = .r, AB = h, and PQ, = 'iy. Tiien the quantity discharged througli the orifice in an element of time, from (9) of Art. 8G, is D B C Fig. 44 Q = p Va^^/yA/^+Vi' -« dx\ dty (1) the .r-integration being taken between h — h' and 0, z being constant during this integration ; and since, in the same time, the surface of the ]i(|uid at 11 descends a dis- tance dz, the quantity discharged through the orifice in this time must equal Kdz, where A' is the area of the sec- tion of the vessel at H. Hence, we have 'HV^fyVx -\- h' — z dx dt = Kdz\ "ly/'igJ fyy/x + h' ~ z dx the 2;-intc^^ration being taken between and h. (^0 (3) EXAMPLE. Find the time of emptying a cone by an orifice ACB in its side. Let AH = h be the axis of the cone, CD = h, CA = /, angle HAC = «, AK — .r, PK being perpendicular to AH. When the orifice is opened, let the surface of the liquid in the vessel be at H, and at the end of the time t let it be at M, and let AM = z. Hr^° EXAMPLE. 167 lirougli A IE B C jig- 44 (1) ^ 0, in the s a (lis- ifice in he sec- (y) Then we have AP = X sec «, and y = PP' = Pp = sec €t dXy h sec a I x\ ,\ the area of Wp'p = — - — x dx. The velocity of discharge through this area therefore the quantity discharged in an element of time = — Y~ V^^ / x'^z ^xdx dt the a;-limits being and z; and this must equal KdZy from (2). Hence we have, from (3), taking the negative sign, be- cause z decreases as t increases, - Kdz t = / b s ecret "T" V^^zi = / — 15/77 tan^* ft z^ dz ^hy/%g sec* « 2^ ISZtt tan^ce /*f?2 4iV25r sec' between the limits h and ;«;. II ■I 158 EFFLUX FROM A VESSEL IX MOTIOX. Therefore the whole time of emptying the vessel 15 ' 'jur« V^ (See J 'ncrB Hydrostatics, p. 185.) 88. Efflux from a Vessel in Motion.— If the vessel ABCD be filled with liquid to AB, and raised vertically, with an accelerated motion, by a weight P attached to an iixextensible string, without weight, passing over two smooth pulleys F, K, the veloc- ity of efflux is augmcMted ; and if it descends with an accebrated motion, the velocity is diminisiied. Let Q be the weight of the vessel and liquid containeJ in it. Since the pulleys are perfectly smooth, the tension of the strir ; is the same tliroughout ; hence the force wiiich causes the motion is the difference between the wei^dits P and Q. The moving force, therefore, is P — Q\ but the weight of the mass moved is P -\- Q. Hence, from (I) of Art. 'Ih, Anal. Mechs., we have J - p+ q'J> (1) which is the vertical force of ace 'oration. Since this force acts vertically uj)wards on the vessel, and the force of grav- ity g acts vertically downwards, every particle of the liquid presses against the bottom of the vessel, not only with its own weight Mg, but also with its inertia M/. Hence the I EFFLUX FROM A VrSSEL IN MOTION, 159 -i 'S f) lllv, ^B entire accelerating force pressing against every point in the ba:e is 9+f=0-^^j^n9 [from(l)] P+ Q 2P P^Q^' (2) Lot 110 = //, and v = the velocit^'^ of efflux ; thop e have (3) v= s/'l(g +f)h + Q (4) Cor. 1. — If the vessel is allowed to empty itself through the orifice 0, without receiving any liquid, let x = the variable altitude OH, A' the horizontal section of the vessel, which is a function of x, and k the section of the orifice. Then we have Q = fKdx; which in (4) gives for the quantity discharged in an element of time. >'^-^!l\/ p~% dt =■ — K dx\ = / K dx ~ Kdxy/TjTfWdx Jcy/Ig- V2Px Coil. 2. — If f = g, (3) becomes V = V^^gh = 2Vgh; ' (5) . and the velocity of efflux is 1.414 times as great as it would be if the vessel stood still. 160 EFFLUX FHOH A ttOTATlXn VKffSKL. 4 COK. 3.— If in a), /* = Q, tljon /' = 0, mid the vessel is at rest. If /* < (J, tlioii (J will (U'scrud and 7' ascend, / is negative and (3) beeonies and the vessel descends with an accelerated motion, the velocity being diminished. Cor. 4.— If P = o, then, from (2),//+/=0, and thfciciore, from (3), r = 0, and there is no pressure on the bottom of the vessel, and no liquid will flow out; which is also evident from this, that every i)arliole in tne vessel will descend by its own gravity, with the same velocity. 80. Ktfliix from a Rotating Vensel. — If a vessel ABCD, containing a liquid, is made to rotate about its ver- tical axis XX', the surface of the licjuid will take the form of a ])aral)oloid of revo- lution (Art. 21), and at the centre II of the bottom the dej)th of liquid KII is less than it is near the edge, and the li(juid will tiow more slowly through an orifice at the centre than through any other orifice of the same size in the bottom. Let h denote the height KH ; then the velocitv of efflux through an orifice at II = V''igfi. Let y denote the distance II O = MP of an orifice from the axis XX', and w the anguli'.r velocity ; then, since the subtangent MT is bisected at K, we have, for the height of the liquid at P above the centre K, KM = iTM = IMP tan MPT MP ■~ *^ MN y2(j2 r ^9 [from (3) of Art. 21]. Vessel Wmi, tho and till. Ji is will r.B THE CLEPSYDRA, OH WATEH-CLOCK. 161 Hence, the velocity of efflux tluough the orifice at is (1) ScH.— This formuhi is true for a vessel of any shape, even when it is closed at the top so that the paraboloid AKB cannot be completely formed. In this case also, // is the depth of the orifice below the vertex K, and yw is the veloc- ity of rotation of the orifice. (See Weisbach's Mechs., p. 819.) ?Hi. The Clepsydra, or Water-Clock.— This is an instrument consisting merely of a vessel from which the water is allowed to escape through an orifice in the bottom, and the intervals of time are measured by the depressions of the upper surface. Thus, if we wish the clock to run 12 hour's, we let ^ = 12 hours = 12 x CO x CO seconds; then solving (4) of Art. 71) for //, we have and substituting in it this value of /, we have A = ^-^(12xOOxCO)2, which gives the depth of lit{uid in the cylindrical vessel that will empty itself in 12 hours. (1) To discover the manner in which the height h of the vessel must be divided in order that the upper surface of the lic^uid may descend through the several divisions of the scale in equal intervals of time, we make / in (1) successively equal to J 2, 11, 10, 4, 3, 2, 1 hours, and get for h a series of values which are as 144, 121, 100, . ... 16, 9, 4, 1 ; hence, if the height h be divided into 144 equal spaces, and ' 162 TUB VENA COATRACTA, murkiMl upwards from tlie bottom of the vessel, then tlie marks Vil, 100, 10, 0, 4, 1, 0, will give the water level at 1, 2, 8, 9, 10, 11, 12 hours after the water begins to flow. {2) Any vessel may serve for a clei)8ydra, but that form is most convenient in which the upper surface of the liquid deseends uniformly. • * Let X = the height of tlie liquid in the vessel, K the area of the descending surface, v its velocity, and k the area of tiie orifice. Tiien from (a) of Art. 78, we have V = ^V2gx, (2) And since the surface is to descend uniformly, this value of V must be e(jual to some constant a, which will de|)end upon the whole height and tlie time in which the clepsydra will be emptied ; hence (2) becomes A'2 = _ ^2(/x^ a* (3) and supposing the area of the descending surface of the liquid to be a circle = n?/^, (3) becomes A^ = 7Ty = -^; 2k^g (4) which is a parabola of the fourth order. Hence, the heights of the sections must vary as the fourth power of their radii. 91. The Vena Con tr acta.— The laws of efflux that have been deduced are founded on the hypothesis that the liquid particles descend in straigiit lines to the orifice, and all is3ue in parallel Une? with a velocity due to the height COEFF/CIhWT OF COyTRAI'TlON. 163 the level \s to fhe irea (2) itie nd ira [3) of the li({uid surface. Experiment shows, liowever, that this is not the case. The liquid does not issue in tlie form of a prism, and hence the quantity discharged in a unit of time is not measured by tlie contents of a i)rism whose base is tlic orifice and whose altitude is the velocity; this would give the theoretical discharge (Art. 7G, C'or. 3), but the prac- tical discharge is geneniUy niucii less. When a vessel enij)tie8 itself through an orifice, it is observed that the particles of liquid near the top descend in vertical lines ; but when they approaci» the bottom they take u curvilinear course, being turned in towards the orifice, or spirally around it, and this deviation fnnn a vertical rectilinear path is the greater the further the horizontal distance of the particles is from the orifice. The obli(|ue direction of the exterior particles within the vessel continues through the orifice, and gives the stream of lijjuid, in issuing from tlie orifice, nearly the form of a truncated cone or i>yramid, whoso larger base is the area of the orifice. This diminu- tion in the size of the issuing stream is called the contrac- tion of the vein, and the section of the stream at the point of greatest contraction is called the Vena Contracta,* or contracted vein. From I results of most experiments, the vena vontractay when the orifice is a circle, is at a distance from the orifice equal to the radius of the orifice. 92. Coefficient of Contraction. — When water flows through orifices in thin plates, it has been found, b.' meas- urements of the stream, made by different experin'.L^nters, that its diameter at the vena conlracta is about O.H jf the diameter of the orifice. The ratio, therefore, of the cross- seciion of the vena contracta to that of the orifice in a thin • ThU name was first given by Newton, who also showed that, hy taking the area of the vena contracta as the area of the orifice, and repardlng the height of the surface above the vena cotUraeta as the height of the vessel, the theoretic discharge agreed fv more cloaeljr with the pr^tical. 164 VOKFFH'IK.\T OF KFFLVX, plato is 0.r»4. This ruiio is ouUcd tho Coefficient of Contrac- tion.* Denoting it by «, wo have nk for tho section of the vena co7ifrar/n, k being Iho section of the orifice (Art. H\). Hubstitntinp nk for k in (0) of Art. 7C, we have, for tiie . 1>3. <'»<^fli('if the theoretical one. This loss of ve- locity arises from the friction of the water n|K)n the inner RurfacM* of the oriticM*, and from the visc^osity of the water. The ratio of the actual velocity to the theoretical velocity is called tie Coefficient of Vehcity. This coeflicicnt is found to bo tolerably constant for dilTerent heads with well-formed simple orifices, and it very often has the value 0.{)7. De- noting the coefticient of velocity by <^, and the actual velocity by r,, we have which is tli(^ a»-tnal velocitv of eftlux. (I) »4. (WHIci,, Q^ = nk(i>t) = akilty/'iyli (1) * TIiIh nitin In not roiiHtant. Itiit iitiilcrKOi')' vnriittioiiH by vKryliiK the forin of IId' oriflcr, thu (hlckiii)HH ors. <* Hurriu-*< in wliicli the orlflcr |m r.iadi\ or Uiu form of Hit voaiwl. f SzpvrimoiitH madi- by Miclidlotti, KyU'lwoin, and otbon. ffrnr- I- ilw (i) KFFLIX TtlliOrHH SHOUT TIUKS, i. cjin alno Iw tlclcrmiiicd hy direct niejwureineiit of the dischar«(e in a jj;iven time, an ohserva- tion wiiieh can Ik? made with nineh greater aeenracy tlian thoHe of contraction and velocity, (»ii which it depends. Tn th«' present vhhi^ it is found hy direct measurement to he Xt'l, agreeing well with the product .04 x .1)7, of th«' values ahovo givt u.* IIkm. — U«|M'nio(l ohHcrvatioMH nnd oxiM>rii>H'iUH liiivo 1«mI to tlio con- nliiHioii tliat tlin <'«M>IIin««nt, of ollliix is not coiiHtniit for nil oritirt'H in thin )>lntrH ; iliiit it in ^n'liti-r for Hninll orifln>H luul Hniiill vi>lo<>iti«>H of ofTlux tlian for liir^o oriliiTH iin forniH arr n*gn- lar or circulKr. For pipno'i' orilhTH, whow ari'aH ar«« from 1 to 1) Hf|iiiin> inclioH, iinilor n head of from 7 to 21 frrt, a«>ntr(ling tothi>rx|Mf1tux \h ii .010, for rirnilar orifh'CH from ^ to (( inclirH in dianifUr, witli from 4 to 20 fort head of wntrr, it ih // .015. or alntnt ^\.^ 115. Kllliix fliroiiKh Short Tubes, or Ajiita^cM. irtlH< water, instead of flowiiiij: throtigh nn orilice in a thin plate, he allowed to dischargf through s/iorf hihrs, ralh'd also n/ufn(/«'s nud inouni-pims, tho «piantity dischargetl frotn a given orilii-c is consirahly iiuToascfl. More seems to he gained hy the adhesion of the liipiid particle,*^ to the !-i. 894 ; also, Tatn'a Moch. Phil., p. 166 EFFLl'X THROUGH SHORT TUBES, OR AJUTAGES. i different degrees of udvantiige in this respect, which can )k3 determined only by experiment. The discharge is fonnd to be greater ^^hen the ajntage is conical and the larger e*ul is t'ne discharging orifice. (1 , The results of many experiments* made with cylin- drical tubes IJ^ to 3 inches in diameter, the length of wiiicli d«>es not exceed 4 times the diameter, as in Fig. 48, and under a head of water varying from '6 to 20 feet, give as a mean value of the coefiicient of efflux, « = .815, or about \. Since the coefficient of efflux for a sim- ple orifice in a thin plate (Art. 94) is ft = .01'), it follows that, wiien the otiier circum- stances are the same, the discharge through .815 a short cylindrical tube = -- - = 1.325 times the discliarge •^ .615 ^ through a simple orifice in a thin plate. These coefficients increase a little when the diameter of tiic tube becomes greater, and decrease a little when the head of water or the velocity of efflux increases. In this tube, the contraction of the stream takes i)lace at the inlet ab, and not at the outlet. If a small hole were bored in the tube at a or h, no water would run out, but air would be sucked in ; and when the hole is enlarged, or v, lien several of them are made, the discharge with a filled tube ceases. Also, if a tube be placed in a vessel of water A, and inserted in the hole at b, the water will rise in the tube Kb, and run out of the tube abed. (2) With a compound mouth-piece, having an enlargement at its exterior orifice or out- let, as well as at its interior orifice, as in Fig. 49, the results of careful experiments f give the coefficient of efflux /* = 1.5526, when tlu^ narrow part cil is treated as the orifice, thus Fig. 49 f * EzperlmeaU nutde by Mlcbelotti. t Mlkdu by Eytclwoiu, COEFFICIENT OF RESISTAXCE. 167 'Hi ffo Id lli giving ft discharge greater than that which is due to the sec- tion cd of tho pipe. Since fi = .015 for a simple orifice, it follows that the diseliargc through the compound mouth- piece 1.5526 = 2.5 times the discharge through a simple orifice in a thin plate, .615 and -:— = l.H times the discharge through u .815 short cylindrical tube. In the experiments made by Eytelwein, the interior diam- eter (lb was about 1.2 times the diameter cd, and the sides oU and dk made with each other an angle of 5° 9'. 1>(>. Coefficient of Resistance. — AVhen water flows from a cistern through a tube kept constantly full, it fol- lows that the coefficient of contraction of tiiis mouth-piece €t = unity, and that its coefficient of velocity = tiie mciiii «>f .!><; and .09 (Art. 03), = 0.975, and therefore we have, from ((J), UOEPFIVIENT OF RESISTANCE. 169 [4) per '^ = [(.4)- = '*■"''' which in (4) gives for the loss of energy, or stored work lost, 0.052 - W, or 5.2 per cent. (8) CoK. 3. — For efflux through a short cylindrical tulie [Art. 95, (1)], we have = .815, since = ^i, and therefore we have, from (G), " = [(.sVs) - '] = «-^»'' which in (4) gives, for the loss of energy, 0.505 ~ W, 2g (9) or nearly 10 times as mucii as for efflux through an orifice in a thin plate. Sen. — Hence, if the kinetic energy of the watt" to be made ut") of, it is better to allow it to flow throu; an ori- fice in a thin plate than through a short cylind' al tube. But if the edge of the tube be rounded off where .: is united to the interior surface of the vessel, and shu hJ like the contracted vein, we have fi = = --L = 0.845 ; V 1-4 hence, from (1) of Art. 9.3, we have, for tlie actual veloc- ity Vi, • a c\ the I = 0.845a/G4.4x3 = 0.845 X 8.025\/3 = 1 1.745 feet ; k = i^Yn = 0.02182 8(|uare feet; hene. the required disciiurgo, from (1) of Art. 04 (since ft = 1) is Q = k(t>V'igh = 0.02182 X 11.745 = 0.25G cubic feet. ' 1)7. Resistance and Pressure of Fluids. — (l) By the resistance of fluids is meant tliat force by which tlie motions of bodies therein are inn)eded. 'Die resistance of a fluid to the motion of a body is occasioned by the force necessary to dis])liicc that fluid. Since the motion commu- nicated to a body at rest by another body inipinging on it with a certain velocity is e(|ual to the motion lost by the impinf,nng body, therefore the motion communicated to the disphiced fluid must be the same as that of the movinj:? body ; licnce tlie work xWwh the fluid destroys in the mov- in^^ body will be e(|U!il (o the work stored iii the fluid. Let a = the area of the front of the body presented to the fluid, = the velocity of the body, to = the weight of dr se t^ I RESrSTANCE ASD PltESSURE OF FLrmS. in feet of )C- a cubic foot of [he fluid, A' = the resistance of the fluid to the motion of ttie body. Then, weight of the displaced fluid per second = avw. But this maes has a velocity of v feet given to it* .'. work geuurated per second in displacing tliis fluid (1) But this work is performed by means of a force which drags the l)ody through the water at the rate f»r . feet per second, against an ecjual and opposite resistance A' - li X V '^9 ' ,, awiP- (2) tliat is, the rcfiistniirc raries as the Sf/uarc of the re- On account of eddies which are fornu-d round the corners of the body and in the rear, I he vaUie of Ji in {'Z) should be multiplied i)y a constiuit /•, giving U = kair ~ RE^f. — The constant k is to He determined by experiment for each form of solid. For a IhhIv whose transverse section is circular, k does not differ much from unity ; for a flat plate moving flat-wise, it is about \."l'). Kcsistances of this kind, however, an- very irregular, and may vary considera bly in the course of tin' same exi)erimo!>t. Dilferent results are therefore obtained by different experimentalists.* * Sec Rankloc'a Applied Mecbs., p. 506 ; aluo Cottcrill'e Applied Mcche., p. 479. 172 WOUK AND PRESSURE OF A STREAM OF WATER, (2) Tho pressure of a current, upon u pluiie is c<|ual to the resistunce suffered by tlie ;) Ix^eomes /' = (7) VjOR. 2. — lA't If = the section of the pipe, and v = the velocity due to the head of waiter /< ; then 11' = ii'Z.bav, which in (7) gives y> = ii'i.bn X U. (8) EXAMPLE. To find the work of a stream of water issuing from a nozzle with a given velocity. Let V be the given velocity, a the area of the nozzle, and w the weight of u cubic foot of water. 'J'heti the weight of 174 IMI'ACT AOAiyST AM' SURFACE OF UEVOLVTION, tlie wiitcr ])rojecU3d work jK?r Rccond IHjr second = awv, mul theroforc (he (1) that is, thr work' rarim an thr ciihr of fhc vcloeUn (tf tin water Vow. — Iji't = the coellick'ut of velocity; then, from (1) of Art. !»){, we liuvu V'i(jli wliich in (1) jifivcs work j)cr seeond = (iMtrhy/'ifjh. C^) 09. Iiiipact of a StiMMiiii of Water against aii.v Surface of Kevoliitioii. — I^'t MA(! he u surface of revo- hition, against wiiich a stream of water KA, moving in the direction ;»f the axis W of the surface, im- pinges. Let ir he the weight of water disciuirged on thi- surface per second, v its veK)city, ?', tlie veloc- ity of tlie surface, and n tiic angle IVn» which tlie tangent IIT to the surface at B makes with the axis AP, or wliich each lihimenl 11 li of the stream of water, on leaving tiie surface, makes witli the direction of the axis HI). Then the water impinges upon the surface with the velocity V — /', ; and, if friction he negU'cted, the water passes over the surfave witli thai velocity, and leaves it in a tangential direction, Til, TK, etc., with the same velocitv. From the tangent iiil Vi'locity lUl = /• — /•,, jind tlie velocity HI) = /', of the surface paralli'l to the axis, we have the result ant velo«-ity HM = V of tlu water, after it has im| on the surface, hy the formula for the parallclogri velocities, I of ani 0) (0 IMJ'A ( 'T A a AjyST A .V K SUKFA CE OF liKVOLt TIOS. 1 75 V = V{ii — v^y -{■ !',» -f 5i {V — «',) f, I'os »t. (I) Now tlu» kinetic energy of the water Iwfore impact = f "'' aiiiil the kinetic energy remaining in the water after imjiaet henee, the kinrtii; energy tran«mitt»'«l !<» the surface = <'"-'■■•■) V (8) If /* Iw put for th<' foree or impulse against the surfsiee, then tiie energy triinsmittetl to the surface = /'<»,, which in (:{) gives /v, =(^-r»);^ ■ = [i^-(r-r,y- r^i-'i{o- /•,) r, cos w] , from (1), II' = (I - co8«) (/- r,)r, ; (4) \y .: r =z {I — COS «) (y - r,) , ||) wliich is the foree of tlu; water against tiie surface in the ilirection of the axis. That is, thr inipidse rnrics as the rclatirc relocity of the irate r. Con. 1. — If the surface moves with a vehx'ity r, in the opiumtc (lireeliim l<» that t>f tiie water, we have, from (5). /' = (1 — cos «) (/• 4- /',) W m ,^^ ^, ^ 1^ N^ ,b> IMAGE EVALUATION TEST TARGET (MT-3) v.- 4&0 ^< w ^ 1.0 I.I ^ lio 12.0 ■ 2.2 u& IL25 HI 1.4 1.6 '/ Photographic Sciences Corporatioii \ <> 4 23 west MAIN STRliT WiBSTiii,N.Y. U5M (716)172-4503 6^ 176 IMPACT AOAINST ANY SURFACE OF REVOLUTION, If the surface is at rest, Vj = 0, and (6) becomes W P = (1 — cos «) V — • SI Cor. 2. — If a = the area of the cross-section of the stream, and w = the weight of a cubic foot of the water, the weight of the impinging water per second is Avhich in (5) and (6) gives aw P = (1 — cos «) {v T v^)^— , if aw and in (7) gives P = (1 — cos «) v^ — J8) (9) (10) Tliat is, the impulse varies as the square of the rela- tive velocity of the water, and also as the area of the cross-section of the stream. Cor, 3. — The impulse of the same stream of water de- pends principally upon the angle « at which the water moves off from the axis after the impact. If the surface BAC is hollow, as in Fig. 52, the water after impact leaves the sur- face in a direction opposite to that in which it strikes it, and thus much more work is done on the body with a surface concave to tiie stream than on one convex to the stream, since the work remaining in the water on leaving the former surface will be less than it is in the water on leaving the latter. If « = 180°, we have cos a — — 1, which in (5) and (6) gives P = %{vTv,)^, (11) Fig. 52 IMP A CT A OAtNST ANT SVRFA CE OF RE VOL UTIOX. 1 71 and in (7) gives P = 2v W (12) (8) Cor. 4. — When the surface is plane, as in Fig. 60, « = 90° and cos « = 0. Substituting this value in (5), (6), and (7), they become W which agrees with (6) of Art. 98 ; and W v^ P z= V — = - aw, from (8), 9 9 = 2 X H- X «^^ = 2hxaw; 2g (13) (14) that is, the normal impulse of water against a plane surface is equal to the weight of a colum^n of water whose base is equal to the cross-section of the stream, and whose height is twice the head of water to which the velocity is due. •% Cor. 5. — If the plane surface (Fig. 50) against which the stream impinges moves away with a velocity w in a direction which makes an angle with the original direction of the stream, the velocity of the surface in the direction of the impact is ^1 = II cos 0, which in (13) gives for the impulce, W P = {v — u cos 0) — , if and the work done by it per second is W Pv^ = {v — ti cos 0) u cos — • 2f (15) (16) ■n 178 ODLIQUR IMPACT. 100. Oblique Impact. — When a stream impinges obliquely on a plane, there are several cases, viz., when the water after impact flows oflE in one, two, or in more directions. (1) Let the plane AB, upon which the stream AC impinges, have a border upon three sides so that the water can flow off in one direction only. Then the impulse of the water against the surface in the direction of the stream is, from (5) of Art. 99, Fig. 53 P = (I — COS n) {v — ?'i) W d (1) Fig. 54 (3) Let the plane AB, upon which the stream DC impinges, have a border upon two sides only, so that the water can flow off in only two directions. The stream will divide itself into two unequal parts, the greater part flowing off in the direction CB, and the other in the direction CA. Let W^ be the weight of the former, W^ the weight of the latter, and W the whole weight. Then the total impiilse in the direction of the stream, from (5) of Art. 99, is w w P = (1 — COS a) {v — i\) — - 4- (1 -f cos a) {v — Vi) — *- = (^ -^) [(1 - cos «) Ifj + (1 + cos a) Tfg]. (2) But the cond tions of equilibrium of the two portions of the stream require that the pressures on CB and CA shall be equal to each other ; hence (1 — COS «) If, = (1 + cos a) Ifg, or, (1 — cos «) TTj = (1 + cos a){W— W^), OBLIQUE IMPACT, 179 from which we find TTj = ^ (1 + cos a) W, and W^=\{l-QOiia)W. Substituting these values of W^ and W^ in (3), we have ^ V — V 9 - W sin'^ «, (3) wliich is the total impulse in the direction of the stream. . . , . Dividing (3) by sin «, we obtain V — V CR = P cosec « =. ivhich is the normal impulse. Multiplying (3) by cot a, we obtain - W sin «, (4) CS = P cot « = V V, 9 V — V W sin €t cos « which is the lateral impulse. i irsin 3«, (5| Hence, ^Tie ^o^ot^ impulse in the direction of the stream is proportional to the square if the sine of the angle of incidence, the normal impulse to the sine of this angle, and the lateral impulse to the sine of double this angle. .r-j-i.j, .. ScH. — If the oblique plane has no border, the water can flow oflE in all directions ; in this case the impulse is in- creased, for a is the smallest angle which the filaments of water can make with the axis, and hence every filament which does not flow off in the normal plane will make with the axis an angle larger than a, and therefore from (3) will exert a greater pressure than those which do. 180 MAXIMUM WORK DONE BY THE IMPULSE. i 101. Maximum Work done by the Impulse.— The work done by the impulse P, from (4) of Art. 99, is Pv, = (1 — cos a) {v — v^) v^ W (1) The work is zero when the velocity of the surface Vj := 0, and also when it = v. To find the value of Vy which makes this work a maxi- mum, we must equate to zero its derivative with respect to t?i, which gives V 3i;, = 0, or v, = ^v, hence, the work done by the impulse is a maximum when the surfaee moves in the direction of the stream, with half the velocity of the stream. Substituting in (1) for v^ its value, we have Pv, =(l-cos«)||Tr, (3) which is the maximum work done by the impulse. Cor. — If the surface is a plane, as in Fig. 50, « = 90°, and we have, from (2), 1 v* Pv — - — W (3) That is, the water transmits to the surface, in this case, one-half of its kinetic energy. If the surface is hollow, as in Fig. 52, so that the water is reversed, « = 180°, and we have, from (2), (4) In this case, the water transmits to the surface all of its kinetic energy. (See Weisbach's Mechs., p. 1010.) HampLes. 181 EXAMPLES. 1. With what velocity will water issue from a small ori- fice 64| feet below the surface of the liquid ? : - ' Ans. 64^ feet. 2. If 252 cubic inches of water flow in one second through an opening of 6 square inches, find the head of water. Ans. 2.28 inches. 3. If water flows from a vessel whose cross-section is 60 square inches, through a circular orifice in the bottom 5 inches in diameter under a head of water of 24 feet, find its velocity. Ans. 41.58. 4. A vessel, formed by the revolution of a semi-cubical parabola about its axis, which is vertical, is filled with water till the radius of its surface is equal to its height above the vertex. Find the time of emptying the vessel through a small orifice at the vertex. [Let ai^ = a:^ be the equation of the generating curve, and ^ the area of the orifice.] 7r«'* /2a Ans. -iiprX/ — Iky g 5. A conical vessel, the radius of whose base is r and alti- tude h, is filled with water; the axis is vertical and the water issues through an orifice in the vertex, of area k. Find (1) the time in which the surface of the water will descend through one-half its altitude, and (2) the time in which the cone will empty itself. Ans. 6. Find the time in which the cone empty itself through an orifice in its base. Ans. 2m lie in Ex. 5 would 16Trr3 15^ 7. A sphere is filled with water. Find the time of empty- ing it through an orifice in its bottom. IGrrr' /r Ans, 182 EXAMPLES. 8. A hemisphere is filled with water. Find the time of emptying it (1) through an orifice in its vertex, and (2) through an orifice in its base. Ans. (1) y^; (2)-^_. 9. A rectangular orifice is 3 feet wide and 1^ feet high, and the lower edge is 2f feet below the level of the water. Find the quantity discharged in 1 second. Ans. 43.7 cubic feet. 10. An orifice in the form of an isosceles triangle, with its vertex in the surface of the water has a base of 1 foot which is horizontal, and an altitude of 6 inches. Find the quantity discharged in 1 second. Ans. 1.135 cubic feet. 11. If the orifice in Ex. 10 has its vertex downwards and its base 6 inches below the surface of the water, and hori- zontal, find the quantity discharged in 1 second. Ans. 1.633 cubic feet. 13. If a vessel, when filled with water to the depth of 4 feet, weighs 350 lbs., and if it be drawn upwards by a weight P of 450 lbs., as in Fig. 46, find the velocity of efflux through an orifice in the bottom. Ans. 17.02 feet. 13. If the vessel (Fig. 47), which is filled with water, makes 100 revolutions per minute, and if the orifice is 2 feet below the surface of the water at the centre, and at a distance of 3 feet from the axis XX', find the velocity of efflux. Ans. 33.4 feet. 14. Find the times in which the surface of water con- tained in a vessel, formed by the revolution of the curve y^ = a^x about the axis of x, will descend through equal distances A, the water issuing through a small orifice in the vertex, and the axis vertical. -na^h Ans. -—:' W2g 15. Water issues through a small orifice 16^*5 feet below the surface of the liquid. If the area of the orifice is 0.1 of EXAMPLES. 183 :«^ a square foot and the coefficient of efflux is 0.015, how many cubic feet of water will be discharged per minute ? Ans. 118.695. 16. A basin has in it u hole an inch square ; water in the basin is kept at a constant level of 9 feet above the hole. How many cubic feet of water will flow out in 1 hour, the coefficient of efflux being 0.6 ? Ans. 360. 17. A cylindrical vessel filled with water is 4 feet high and 1 square foot in cross-section, and a hole of 1 square inch is made in the bottom. If the coefficient of efflux is 0.6, in what time will f of the water be discharged ? Ans. 00 seconds, nearly. 18. A cylinder, the urea of whose cross-section is 60 sq. ft., is filled with water to a depth of 12 feet ; a small hole is made in its bottom, whose area is 0.5 square inches. In how long a time will the depth of the water be (1) 8 feet and (2) 4 feet? Ans. (1) 45.8 minutes; (2) 105.4 minutes. 19. The horizontal section of a cylindrical vessel is 100 square inches, its altitude is 30 inches, and the area of its orifice is 0.1 of a square inch. If filled with water, in what time will it empty itself, the coefficient of efflux being 0.62 ? Ans. 11 m. 36.5 s. 20. What is the discharge per second through a rectangu- lar orifice 2 feet wide and 1 foot high, when the surface of the water is 15 feet above the upper edge, the coefficient of efflux being 0.611 ? Ans. 38.6 cubic feet. 21. What is the discharge per second through a rectan- gular orifice whose height is 8 inches and whose width is 2 inches, under a head of Avater of 15 inches above the upper edge, the coefficient of efflux being 0.628 ? Ans. 0.705 cubic feet. 22. If the height of the rectangular orifice is 15 inches, its width 25 inches, and the head of water is 4^ inches above the upper edge, what is the discharge per second, the coefficient of efflux being 0.594 ? Ans. 12.19 cubic feet. 184 SXAMPLSS. 23. A plane area moves perpendicularly through water in which it is deeply imbedded. Find the resistance per square foot at a speed of 10 miles an hour. Ans. 269 lbs. 24. A stream of water delivering 100 cubic feet per min- ute, at a velocity of 15 feet per second, strikes an indefinite plane normally. Find the pressure on the plane. Ans. 48. G lbs. 25. If a stream of water, the area of whose cross-section is 64 square inches, impinges with a velocity of 40 feet per second against the convex surface of an immovable cone, in the direction of its axis, the vertical angle of the cone being 100°, find the impulse. Ans. 492.16 lbs. 26. A stream of water, the area of whose cross-section is 40 square inches, delivers 5 cubic feet per second, and strikes normally against a plane surface, which moves away with a velocity of 12 feet per second. Find (1) the impulse, (2) the maximum work, and (3) the maximum impulse. Ans. (1) 58.125 lbs.; (2) 784.688 ft. -lbs.; (3) 87.19 lbs. CHAPTER II. MOTION OF WATER IN PIPES AND OPEN CHANNELS. 102. Resistance of Friction. — When a thin plate with sharp edges, completely immersed in water, is moving edgeways through the water, a certain resistance is expe- rienced, which must be overcome by an external force. This resistance acts along tangeutially between the plate and the water, and so far is analogous to the friction be- tween solid surfaces, but it follows quite different laws, which have been obtained from many observations and ex- periments, and which may be stated as follows:* (1) The resistance of friction is entirely independent of the pressure on the surface. (2) It varies as the area of the surface in contact with the water, (3) It varies nearly as the square of the velocity, f Hence, if R be the resistance of friction, S the area of the surface, and v the velocity, these laws may be expressed by the formula, R=fSv^, (1) where /is called the "coeflBcient of friction," as in the fric- tion of solid surface?. The value of / depends on the smoothness of the surface; thus, for thin boards, with a clean, varnished surface, moving through water, it is found • Cotterill's App. Mechs,, p. 468. t At low Telocitie?, of not more than 1 inch per second for water, the resistance varies nearly as the first power of the velocity. At velocities of \ foot per seeond, and greater velocities, the resifltance vAries more nearly ae the square of tbe ve- locity. 186 MOTION OF WATER IN PIPES, I to bo .004, while for ii surface resembling medium sand- paper, it is .009, the utiits being pounds, feet, and seconds.* 103. Motion of Water in Pipes. — When water is conveyed to any considerable distance in pipes, the friction of the internal surface causes u great resistance to tlie flow. By the theoretical rule, the velocity of discharge v would 1)6 due to the vertical depth h through which the water falls (Art. 76) ; but owing to friction, theoretical results are of very little practical value. Besides, the friction is often quite uncertain, the central parts of the stream move more quickly than the parts in immediate contact with the pipe, and, though the circumstances are diflFerent, the velocity over the internal surface is liable to changes, as in the case of solid surfaces. The value of / therefore has to bo ob- tained by special experiments, and the results of such experiments do not always agree with each other. It is found, however, that/ lies between the limits .005 and .01, depending partly on the condition of the internal surlace, and partly on the diameter and velocity; its value being greater in small pipes than in large ones, and greater at low velocities than at high ones. The mean of these limits, or .0075, is sometimes taken for/, when there is no special cause for increased resistance. Let V = the velocity of discharge in feet per second, (I = the diameter of the i)ipe in feet, I = the length of the pipe in feet, h = the head or fall of water in feet, and W = the weight of water in pounds discharged per second. Let /' be the resistance of friction due to a unit of diam- eter, length, and velocity ; then the resistance in a pipe I feet long and d feet diameter with a unit of velocity will be, from (1) of Art. 102, f 'hi; but the quantity of water deliv- • For large pnrfacep, eppocially of considerable length, the friction iH very muth dimlniflhed. For instancp. these valneo of/ were obtained by experimentirK on a enrfficc 4 feet long, moving 10 feet per second : bnt when the length was SO feet and npwardB, these values of/ were diminished to .0OS?5 and .005 respectively. I MOTION OF WATER IN PIPES, 187 • ered by this pipe will be d^ times that delivered by the for- mer, therefore for the same quantity of water delivered as by the former, the resistance of friction in the latter pipe M'ill be fid .._ r,l 'Ss- or # ^' d' that is, the rcslstanccy of friction in pipes, when the ve- locity is constant, varies directly as their lengths aiul inversely as their diameters. If we measure this resistance by a column of water, and denote the height of this column by 7ij, we have 7 -f^"^ '*» -^d2g' (1) where/ is a constant to be determined by experiment, and is called the coefficient of friction. This height h^ is called theheigJit of resistance of friction ^ which has to be subtracted from the total head A, in order to obtain the height necessary to produce the velocity v. Hence, the loss of head or of pressure, in consequence of the friction of the water in the pipe, is found by multiply- ing the head due to the velocity by the coefficient /-%, and is greater, the greater the ratio of the length to the diam- eter and the greater the height due to the velocity. Multiplying (1) by W, we obtain for the work due to the resistance of friction • h^W J d%g^' (2) that is, the loss of work by fHction is the same as that of raising the water through a height h^. CoH. — From (4) of Art. 96, W3 have 188 UNIFORM PIPE CONNECTING TWO RESERVOIRS. loss of work due to the resistance at ingress = (i—W', (3) work stored in the water at discharge = — TT. (4) 104. Uniform Pipe connecting Two Beseryoirs^ when all the Resistances are Considered.— Let h be the difference of level of the reservoirs, and v the velocity, in a pipe of length I and diameter d. Then we have work due to the head of water = h W, (1) which is the whole work done per second in moving W pounds of water from the surface of one reservoir to the surface of the other. This work is equal to the work in overcoming all the resistances, together with the work re- maining in the water at discharge. That is, the work is expended in three ways: (1) The head 5- ,* corresponding to an expenditure o{ ^W foot-pounds of work, is employed if in giving energy of motion to the water, and is ultimately wasted in eddying motions in the lower reservoir. (3) A portion of head /3jr-, corresponding to an expenditure of Px-W foot-pounds of work, is employed in overcoming the resistance at the entrance to the pipe. (3) the headf f-j-^f corresponding to an expenditure of f-^jrW foot- d iff U /iff pounds of work, is employed in overcoming the surface friction of the pipe. Hence, from (1), and (3), (3), (4) of Art. 103, we have .1 v^ v« V2. *'^ = /3^,"' + ^-^'^+2-^'^' * Oftlled vekxMif htai. i WMiJHetliOnhtad. ^ UNIFORM PIPE CONNECTING TWO RESERVOIRS, 189 *=(i+''+4)|' and ^ 1 + J34. + i34-/ V or = »•''* V(rT M i3)rf+/r (2) (3) where the constants j3 and /are to be determined by experi- ment. When V and d are given, (2) is used to determine h\ when h and c? are given, (4) is used to determine v. Cor. 1. — If the pipe is bell-mouthed, (3 is about .0». If the entrance to the pipe is cylindrical, /3 = 0.505. Hence, 1 -f j3 = 1.08 to 1.505. In general, this is so small com- pared with /j that for practical calculations it may be neglected; i, e., the losses of head, except the loss in sur- face friction, are neglected. It is only in short pipes and at high velocities that it is necessary to take account of the term (l+i3). For instance, in pipes for the supply of tur- bines, V is usually limited to 2 feet per second, and the pipe is bell-mouthed. In this case, we have (1 . /3) J? = 1.08 X 4 X .0155 = 0.067 foot.* ^ 2g ... , . ... In pipes for the supply of towns, v may range from 2 to 4| feet per second, and then we have n A_p\^ — 0.1 to 0.5 foot. In either case, this amount of head is small compared • £- = .0166 foot. 190 UA'JFOHM I'IFE CONNECTING TWO RESERVOIRS. with the whole fall in the cases which most commonly occur. Cor. 2. — For very long pipes, 14-/3 is so small compare^ with /^, that it may be neglected altogether, and (3), (3), and (4) become I^qhd V = 8.025a / M fl (5) (6) (7) Using the value of/, as determined by Poncelet, viz.,/ = .023, with the value of /3 = .5, we have, from (3), V 47.0\/ hd I + 54c? (8) Eytelwein gave a formula which nearly coincides with tliis. (See Storrow on Water Works, p. 56.) When the pipe is very long, d is very small compared with ly and (8) becomes . = «y f . (9) Rem. — It is immaterial as regards the velocity, and the quantity discharged, whether the pipe is horizontal or inclined upwards or downwards, so long as the length of the pipe and the total head, or deptli of the end of discliarge below the level of the surface of the water in the reservoir, remain unchanged. If the inclined pipe is longer than tlie liorizontal, of course its surface will present more fric- tion against the motion of the water than the horizontal one, and thus diminish the velov^^'^y of discharge; but if the inclined pipe be the same length as the horizontal, and have the same head, then each of them will discharge the same quantity in the same time. It is evidently necessary, in every case, that the eatrance to the 1 COEFFICIENT OF FRICTION F\.' PIPES. 191 pipe from the reservoir be placed suffidentl j far below the water sur- face of the reservoir to allow the water to flow from the reservoir into the pipe, as fast as it afterwards flows al'^ng or through the length of the pipe to the end of discharge. For there must be at least sufficient head to overcome the resistance at the entrance to the pipe, and to allow the water in the reservoir to flow out of an opening freely into the air with that velocity which previous calculation shows it will have in the pipe. The remainder of the head, which is employed in overcoming the resistance of friction, and perhaps other resistances which will be considered hereafter, may be obtained by having the pipe incline downwards. Since the friction in pipes of the same diameter increases as their lengths, when the water first enters the pipe it is opposed by but little friction, and has great velocity ; but this velocity gradually diminishes ns the advancing water meets the friction along increased lengths of the pipe, and finally becomes least when the water fills the whole length and begins to flow from the end of discharge. The velocity then becomes uniform along the pipe, and will continue to be so, if the velocity head and head due to the resistance at the entrance to the pipe are together sufficient to allow the water of the reservoir to enter the pipe with this same velocity. \ 105. Coeflftcient of Friction for Pipes Discharg- ing Water. — From the average of a great many experi- ments, the value of / for ordinary pipes Is / = 0.030268. (1) But practical experience shows that no single value can be taken applicable to very different cases. The coefficient of friction, like the coefficient of efflux, is not perfectly con- stant. It is greater for low velocities than for high ones, i. e., the resistance of friction of the water in pipes does not increase exactly as the square of the velocity. Prony and Eytelwein assumed that the loss of head by the resistance of friction increases with the first power of the velocity and with its square ; and hence they established for this loss of head the formula h = («xV + a^^) ^, (2) 192 COEFFICIENT OF FRICTION FOR PIPES. in which Wj and a^ denote constants to be determined by experiment. In order to determine these constants, these authors availed themselves of 51 experiments made at different times by Couplet, Bossut, and du Buat upon the flow of water through long pipes. From these 51 experiments, the fol- lowing numerical values were obtained : Prony obtained, «i = 0.0000693, «g = 0.0013932. Eytelwein, «i = 0.0000894, «, = 0.0011213. D'Aubuisson,* «i = 0.0000753, «, = 0.0013700. Taking the value of /tj, and substituting it in (2) of Art. 104, instead of the value of h^ as given in (1) of Art. 103, we have A = (1 4-/3)|+(«ii' + «,!;») ^. (3) Putting —= — = I, and reducing, (3) becomes ^9 hd = hdv^ + «g^6'2 _^ a^lvj (4) from which the value of v may be found. But the follow- ing method of approximating to the value of v is more convenient. From (4) we have \ ^ M-i- a^l v) \ hd-\- a J Expanding the first member by the binomial theorem, and neglecting all the terras of the expansion after the sec- ond, since a^ is considerably greater than «,, we have r a^l \\ _ I hd I \ 4 4|.Wei8bach*s Mechs., p. 868. 1 COEFFICIENT OF FRICTION FOR PIPES. A V=.sJ M «i/ bd + ttg/ 3 {bd -f a^l) 193 (5) Now if the pipe is cylindrical, (i = 0.505, from Cor. 3 of Art. 69, and therefore we have 1 +i3 _ 1.505 = .0234, and taking «i = .00007 and «g = .00042,* and substi- tuting these values in (5) and reducing, we have ' = sj\ 2380hd I / + 54rf 12(l-{-54d) (6) Cor. — When h is not very small, the last term of (6) may be neglected, and we have ^ = \/i 23S0hd in l-\-6^d* ' . wliicli is very nearly the same as (8) of Art. 104. When the pipe is very long, d is very small compared vvitli I, and (6) becomes /2dS0hd 1 (8) When d is expressed in inches and all the other dimen- sions in feet, (8) of Art. 104 becomes 2M 5d* (9) ScH. — The following short table gives Weisbach*s values of the coefficient of friction for different velocities in feet per second : f * Tate'B Mech. Phtl., p. 398. t Ency. firit., Art. HyarofMchai^. 194 THE QUANTITT DISCBAROED FROM PIPES, 4' __ / = 0.1 .0686 1 .0315 0.2 .0527 0.3 .0457 0.4 .0415 2 .0265 0.5 .0387 3 .0243 0.6 .0365 0.7 .0349 6 .0214 0.8 .0336 0.9 .0325 .0297 .0284 4 .0230 8 .0205 12 .0193 EXAMPLE. The length of a water-pipe is 5780 feet, the head of water is 170 feet, and the diameter of the pipe is 6 inches. Ke- quired the velocity of discharge. 2380 X 170 X .5 5780 By (6), we have _ 1. ^ ~ y 5780 + 54 X .5 12 (5780 + 54 x .5) By (8), we have /2380 X 170 X .5 1 .., '=S—hm i^ = "-^^- By (8) of Art. 104, we have V = 47.9\/ .,^Q-. , .,' — = = 5.8 feet, nearly. V 5780 + 54 X .5 ' ^ By (9) of Ai't. 104, we have V- 47.9a/ = 5.81. 170 X .5 5780 = 5.8. It will be observed that these results are very nearly the same. 100. The Quantity Discharged from Pipes.— Let Q be the number of cubic feet discharged per second; then ^ is given by the formuL^ 4 THE QUANTITY DISCHARGED FROM PIPES. 195 ^ = -cPy = 0.7854^2^; (1) and on substituting the value of v obtained from (1) of Art. 103, this becomes Q (2) which gives the value of Q in cubic feet per second, since all the dimensions are in feet. If we require the number of gallons discharged per min- ute for a diameter of d inclies, (1) becomes G = ^Vt''*' (3) where C is a constant whose value for / = .03 is 30, but which is often taken somewhat less (say 37), to allow for contingencies.* Assuming that 1 culnc foot = 6.2322 gallons, we have, from (1), for the number of gallons discharged in 24 hours. 7T Q = -- (Pv X 86400 X 6.2322 = 2d'd6Mdh. From (1), we have ,^i|^^l.,,33|. (4) (5) which, in (1) of Art. 103, gives (6) that is, the height of resistance of frictiorv in pipes varies inversely as the fifth power of the diameters, and directly as the length of the pipes. * 8ee Cotterill'fl App. Medw., p- 401. 196 EXAMPLES. Hence, if we wish to conduct a given quantity of water through a pipe with as little loss of head as possible, we must make the pipe as short and its diameter as large as we can. If the diameter of one pipe is double that of another, the friction in the former is -^ of that in the latter. Cor.— Putting 14-/3 = 1.505, and ^ = 0.0155, we have, from (3) and (4) of Art. 104, ^ h = /l.505 +/.^)o.0155t;«, and V = 8.025a/ ^ 1.505 +/• m (8) d EXAMPLES. 1. How many gallons of water would the pipe in the ex- ample of Art. 105 deliver in 24 hours ? Here v = 5.8 and c? = 6 inches ; we have, from (4), Q = 2936.86 x 6^ x 5.8 = 613216 gallons in 24 hours. 2. What must the head of water be, when a set of pipes 150 feet long and 5 inches in diameter is required to deliver 85 cubic feet of water per minute ? Here we have, from (5), 25 12*^ V = 1.2732 X ^ X ^ = 3.056 feet, bO 5* and therefore (Art. 105, Sch.), /= .0243, which in (7) gives 150 X 12> h = (l.505 + .0243 x ^^^^ -) .0155 x 3.056*' = (1.505 + 8.748) .0155 x 9.339 = 1.484 feet. THE DIAMETER OF PIPES, 197 3. Solve Ex. 1 by using the value of v as obtained from (8). ■ From (8), we have ^ 1. 170 505 +/^ Since v is somewhere between 3 and 10, we assume / = .02, and obtain / 170' 505 -h 231.20 = 6.859. But V = 6.9 gives more correctly (Art. 105, Sch.) / — .021, and therefore we have V = 8.025\/^ 170 :505'4- 244.265 = 6.695, which gives the true value to the first decimal place. The discharge, from (4), is Q = 2936.86x62x6.7 = 708370.632 gallons in 24 hours. This result is somewhat larger than that obtained from the value of v in (8) of Art. 104. 107. The Diameter of Pipes.— Substituting in (1) of Art. 106 the value of v given in (9) of Art. 105, we have ^ ^ / 191.27i<^ . 198 EXAMPLE, ' . d — p'^5.8ig»(/4- 4.5 . and therefore from the table c z= 0.606, and from (5),* Hence, the whole head is * Since c cornea between two numbers in the table, /3 ia found more accurately from (5). 204 ELBOWS. therefore, the velocity of efflux is V^gh 8.025VT6 V = - .— 1— = -zz^r^ = 4.61, Vl + i3 V4.74 and consequently the discharge, from (1) of Art. 106, is ^ ... ^, Q = ^d^v = jx4x4.51xl2 4 4 . . = 170 cubic inches. ' Fig. 58 110. Elbows. — When pipes are bent so as to form elbotvs, they present resistances to the motion of water in them ; and these resistances, like many other phenomena of efflux, can be determined only by experiment. If a pipe ACB forms an elbow, the stream sep- arates itself from the inner sur- face of the second branch of the pipe, in consequence of the cen- trifugal force. If the second branch is very short, termi- - . - nating, for instance, at ab, the efflux will be smaller than the full cross-section of the pipe. But if the second branch is longer, terminating at B, an eddy is formed at D, and beyond this the pipe is again filled, so that the velocity of efflux V is less than the velocity at D. This diminution of the velocity of efflux must be treated in the same way as the resistance produced by a contraction in the pipe (Art. 109). Hence, if a is the cross-section of the pipe, and c is the coefficient of contraction, the section of the stream at D is ca, and the velocity v' of the contracted stream is tSLBOWS. 206 and hence the loss of head in passing from D to B is = |3 ^g (1) The coeflBcient of contraction c, and therefore the cor- responding coefficient of resistance P, depends upon the angle of deviation BCE. From experiments with a pipe 1^ inclies in diameter, Weisbach found the coefficient of re- sistance to be a 0.9457 sin* f -f 2.047 sin*|, (2) by which he computed a series of coefficients of resistance for diflferent angles of deviation.* From (2) it follows that the kinetic energy of water in pipes is considerably dimin- ished by elbows. If the elbow is right-angled, we have, from (2), j3 = 0.9846, which in (1) gives h. = 0.9846 i^^ * ^g hence, at a -\ght-angled elbow, very nearly the whole head due to the velocity is lost. ScH. — If to one elbow ACB another elbow is joined, the second one turning the stream to the same side as the first one, there is no further contraction of the stream, and therefore, for efflux with full cross-section, (i is no larger than for a single elbow. But if the second elbow turns the stream to the opposite side, the contraction is a double one, and the coefficient of resistance is consequently twice as great as for a single elbow. ♦ Bncy. iirlt., Vol. XH., p. 487. 206 BENDS. 111. Bends.— When the pipes have curved bends, the resistance is much less than in elbows. If a pipe ACB is curved, it also, in consequence of the centrifugal force, causes the stream to separate itself from the concave surface, and to form a partial contraction. If the bend terminates at BD, the cross- section of the stream at its outlet is smaller than that of the pipe. But if the bend is terminated by a long straight pipe BF, an eddy is formed at D, and beyond this the pipe is again filled, so that the velocity of efflux v is less than the velocity at D. ' If c is the coefficient of contraction, the velocity v' of the contracted stream is V V and hence the loss of head in passing from D to F I'i _ jv' — v y _/!_-, fv^ K 'iy V* -% (1) This is Weisbach's method, but the coefficient of contrac- tion for bends is not very satisfactorily ascertained. If r = the radius of the pipe = MH = HC, and p =. the radius of curvature = HO, then Weisbach's formula for the coefficient of resistance at a bend in a pipe of circu- lar section is ^ = 0.131 + 1.847 ('-T; (8) and for bends with rectangular cross-sections, /} = 0.184 + 3.104 (|-)', (8) V . kqvivaLent pipes. a07 18 where s is the length of the side of the section parallel to the radius of curvature p. (See Weisbach's Mechs., p. 897.) V B 111^/. Pipe of Unifonii Diameter fiqiiivalent to one of Varying Diameter.— Pipes for the supply of towns * often consist of a series of lengths, the diameter for each length being the same, but diflfering from those of the other lengths. In approximate calculations of the head lost in such pipes, it is generally accurate enough to neglect the smaller losses of head and to regard only the friction of the pipe, and then the calculations may be facilitated by reducing the pipe to one of uniform diameter, having the same loss of head. Such a uniform pipe is culled an equiv- alent pipe. Let A be the pipe of varia- ^ ble diameter, and B the ^ .. -- i ' equivalent pipe of uniform diameter. In A let Zj, Zg, etc., be the lengths, d^, d^, etc., the diameters, t\, Vg, the velocities for the successive portions, and let I, d, v, be the corresponding quantities for the equivalent uniform pipe. Then the total loss of head in A due to friction is and in the uniform pipe B, • d'ig If these pipes are equivalent, we have Fig. 59a * Sach pipes are called water maim. 208 DTSCBAROE DIMINISHINO UNIFORMLY. But since the discharge is the same for all portions, cP '^t = ''^2; «'8 = ^Jli'y etc. (3) Then supposing that / is constant for all the pipes, we have, from (1) and (2), I -^Ik. = #( + i + L rfjS ch^ (h^ + etc •). (3) which gives the length of the equivalent uniform pipe which would have the same total loss of iiead, for any given discharge, as the pipe of varying diameter. Cor. — If the lengths of the successive portions are all equal, we liave Zj = Zg = ^3 = etc., and (3) becomes ' = '''''t» + ^» + i + ^S (*) it =IF =^ F 111&. Pipe of Uniform Diameter with Diseliarge Diminishing Uniformly along its Length. — In the case of a branch main, the water is delivered at nearly equal distances to service pipes along the route. Let AB be a main of diameter d and length L ; let Q^ cubic feet per second enter at A., and let q cubic feet per foot of its length be delivered to service pipes. Then at any point C, I feet from A, the discharge is Q ~ Qo — Q^' Consider a short length dl at R The loss of head in that length is Fig. 59h (3) (3) (4) k GENERAL FORMULA FOR ALL THE RESISTANCES. 209 Hence, the whole head lost in the length AB is = ^^w-^^»^-^*^''^'^' (1) or, putting P = qL, the total discharge through the ser- vice pipes between A and B, (1) becomes h = ^^. (^0^ - PQo + in. (2) The discharge at the end B of the pipe is Q^- P-' If the pipe is so long that Q, - P = ^, all the water passes into the service pipes, and (2) becomes ^\_l§/^p2 (3) (See Ency. Brit., Vol. XII., p. 486.) 112. General Formula when all the Resistances to the Flow of Water are Considered.— Let (iy be the coefficient of resistance for enlargements and contrac- tions (Arts. 108 and 109), and d^ the coefficient of resist- ance for elbows and bends (Arts. 110 and 111). Then adding together (3) of Art. 105, (4) of Arts. 108 and 109, (1) of Arts. 110 and 111, we have for the emire head h. A = (1 + ^) ^ + («i^ + «8^)^ + ^1 2^ + ^^2^ = (ccjy -H «8t^) ^ + (1 i- ^ + ^1 " ^«)2^' (1) 210 EXAMPLE. where the values of «j and Wg are given in Art. 105, jflj in Art. 109, jSg in Art. 110, and d = .505 to .08. Neglecting «i, since it is very small compared with «, (Art. 105), and putting / = 'Zga^ = .03, from (1) of Art. 105, wc have, from (1), A = (4 + l + ^ + fl,+/3,)±;. (2) ScH. — An enlargement should be made in the pipe at any considerable bendings ; and when any change takes place in the diameter of the pipe, the parts at the junction should be rounded off. At all considerable bends, where the pipe changes from ascending to descending, a provision should be made for clearing the pipe of the air which is disengaged from the water. Unless some provision is made for the escape of this air, it will accumulate in the highest bends and obstruct the flow of the current. EXAMPLE. ^t In the example of Art. 105 there are 40 bends in the pipe, each having the radius of curvature exceeding ten times the radius of the pipe. Find the velocity of efflux. Here / = .03, (i = .505, i3i = 0, jSg = .131 + 1.847 (tV)^ = .1312; .-. mz = 5.248, I = 5780, d = .6, h = 170. Hence, from (2), we have 170 = (.03 x^ + 1.505 + 5.248) g^ = 353.553 9-2 64i' ,% V = 5.562 feet per second. FLOW OF WATER IN RIVERS AND CANALS. 311 B. in Art. (3) For practical calculations on the flow of water in pipes, see Ency. Brit., Vol. XII., p. 488. 113. Flow of Water in Rivers and Canals.— When water flows in a pipe, the section at any point is de- termined by the form of the boundary. When it flows in an open channel with free upper surface, the section de- pends on the velocity due to the kinetic conditions. Tho bottom of the channel and the two banks are called the bed of the stream. A section of the stream at right angles to the direction in which it is flowing is called a transverse section, and of the line bounding this section, the part that is beneath the Avater surface is called the wetted perimeter. A vertical section in the direction of the stream is called the longitudinal section or profile. Let ABCD represent a longitudinal section of a limited portion of a stream, AD, BC, two transverse sections, AB the surface of the stream, DC the hot tom of the channel, and AE a horizon- tal line. Let I = the length of AB in feet ; h = BE, the difference of level of the water surface in feet at the two extremities of the distance I; z= the angle BAE, the slope of the stream ; sin = -j = the sine of the slope, or the fall of the water surface in one foot ; a .— the area of the transverse section at BC in square feet ; p = the length of the wetted perimeter of the transverse ")tion at BC ; r = -, the hydraulic mean depth, or the mean radius of the section ; Q = the discharge through the section at BC in cubic feet per second ; v = -^ = the a mean velocity of the stream in feet per second, which is tftken as the common velocity of all the particles^ Fig. 60 212 DIFFERENT VELOCTTTES IN A CROSS-SECTION. \\^ 114. Different Velocities in a Cross-Section.— The velocity of tlie water is not uniform in all points of the same transverse section. In all actual streams the different fluid filaments have different velocities. The adhesion of the water to the bed of the channel, and the cohesion of the molecules of water cause the particles of water nearest to th^^ sides and bed of the channel to be most hindered in their motion. For this reason, the velocity is much less at the bottom and sides than it is at the surface and centre. According to some authors, the maximum velocity in a straight river is generally found in the middle of its sur- face, or in that part of the surface where the water is the deepest.* Theoretically we should expect this, but practi- cally it is often very different. The theory adopted bj most modern writers is the fol- lowing: The motion of the water being caused solely by the slope of the surface, the velocity in all parts of any trans- verse section of the river would be equal, were it not for the r»;i,arding influence of the bed. The layer of elementary particles next to the bed adheres firmly to it by virtue of the force of adhesion. The next layer is retarded partly by the cohesion existing between it and the first, partly by the friction, and partly by the loss of kinetic energy arising from constant collision with the irregularities which correspor.d to those of the bed. The next layer is retarded in the same manner, but in a less degree. Thus, according to this theory, the effect of the resistances is diminished as the distance from the bed is increased ; and assuming, as is usually done, that no sensible resistance is experienced from the air, the maximum velocity should be found in the sur- face filament situated at the greatest distance from the bed. The many experiments, however, which have been made to determine the actual variation in velocity at different depths, and upon the surface, at different distances from the banks, give very different results. * Weiebach'8 Mechs., p. 956 ; also Tate's Mech. P\A\., p. 902, DIFFERENT VELOCITIES IN A CROSS-SECTION. 213 Focacci found that in a canal 5 feet deep, the maximum velocity wos from 2 to 3.5 feet l)elow the gurface. Defontaine states that in calm weather the velocity of the Rhine is greatest at the surface. Raucourt made ex|)eriments upon the Neva where it is 000 feet wide and of regular section, the maximum depth being 63 feet. When the river was frozen over, the maximum velocity (2 feet 7 inches per sec- ond) was found a litt.e below the middle of the deepest vertical, where it was nearly double the velocity at the surface and bottom, which were nearly equal to each other. In summer, he found the maximum velocity was near tlio surface in calm weather ; but when a strong wind was blowing up stream, the surface velocity was greatly dimin- ished, so that it hardly exceeded tliat at the bottom. He considers the law of diminution of velocity to be given by the ordinates of an ellipse whose vertex is a little below the bottom, and whose minor axis is a little below the surface. Hennocque found the maximum velocity in the Rhine to be, in calm weather, or with a light wind, \ of the depth below the surface ; in a strong wind up stream, it was a little below mid-depth ; in a strong wind down stream, it was at the surface. Baumgarten found in the Garonne that the maximum velocity was generally at the surface, but that in one section (about 325 feet wide) it was always below the surface. D'Aubuisson considers that the velocity diminishes slowly at first, as the depth increases, but that near the bottom it is more rapid. I'hc bottom velocity, however, is always more than half that of the surface. Boileau found, by experiment in a small canal, that the maximum velocity was J to \ of the depth below the surface. Below this point, the velocity diminished rapidly, and nearly in the ratio of the ordinates of the parabola whose axis was at the surface. He decided, from a discussion of the experiments of Defontaine, Hennocque, and Baum- garten, that in large rivers the maximum velocity is by no means al- ways at the surface.* It will be seen from this synopsis that there is a great diversity among the results obtained by different experi- menters, and that no mathematical relation, of sufficiently general application to constitute a practical law, has been vet discovered. * See r-''>ort on the Hydraulics of the Mississippi River, by Humphreys and Abbott, pp. aoo, etc. 214 niFFitUitw'T vELOcrriEa in a cross-section. The velocities observed on any given longitudinal section, at any given moment, do not form, when plotted, any regular curve. But if a series of observations are talcen at each depth, and tiio results aver- aged, the mean velocities at each depth, wlien plotted, give a regular curve agreeing very fairly with a parabola wiiose axis is horizontal, cor- responding to the position of the filament of maximum velocity. All the best observations show that the maximum velocity is to be found at some distance below the free surface. In the experiments on the Mississippi River, the velocities on any longitudinal section, in calm weather, were found to be represented very fairly by a parabola, the greatest velocity being at -^(^ of the depth of the stream from the surface. With a wind blowing down stream, the surface velocity is increased and the axis of the parabola apprcnches the surface. With a wind blowing up stream, the surface velocity is diminished and the axis of the parabola is lowered, sometimes to half the depth of the stream. The observers on the Mississippi drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream, like that at the bottom and sides. If there were such a retarding action, the position of the filament of max- imum velocity below the surface wouh' '^^ explained. If there were no such resistance, the maximum velocity should be at the sur- face. It Is not difficult to understand that a wind, acting on surface rip- ples, should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the sur- face velocity of a stream Is variable when the mean velocity of the stream is constant. Hence observations of surface velocity, by floats or otherwise, should only be made In very calm weather. But it is very difficult to suppose that, In still air, there Is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. In very careful experiments, Bolleau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. Prof. James Thomson has given a much more probable explanation of the diminution of the velocity at and near the free surface. He points out that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off In eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity In all parts of the stream, but have their greatest Influence at the free surface. Reaching the free L TRANSVERSE SECTION OF THE STREAM. 215 Hurface, thej spread out and remain there, mingling with the water at thai level, and diminishing tlie velocity which would otherwise be found there.* 115. Transverse Section of the Stream. — The form of the transverse section and the direction of the cur- rent have such an effect upon the velocity at tlie surface, at different distances from the banks, that there can be no definite law of change. There is generally an increase of velocity, as the distance from the banks is increased, until the maximum point is reached. That portion of the river where the water has its maximum velocity is called the line of current or axis of the stream, and the deepest portion of the stream is called the mid-channel. When the stream bends, its axis is generally near the concave shore. It is observed that the surface of a stream, in any cross- section, is highest where the velocity is greatest, which is accounted for by the fact that, when the water is in motion, it exerts less pressure at right angles to the direction of its motion than when it is at rest, and therefore, where the velocity is greatest the water must be highest, to balance the pressure at the sides, where the velocity is less. It frequently happens that, while the mass of the water in a river is flowing on down the river, the water next the shore is running up the river. It is no unusual thing to find a swift current and a corresponding fall on one shore doivn stream, and on the opposite shore a visible current and an appreciable fall up stream ; L e., on one side of the river the water is often running rapidly up stream, while on the other side it is running with equal or greater rapidity down stream. The apparent slope at every point is affected by the bends of the river, and by the centrifugal force acquired by the water in sweeping round the curves, and by the eddies which form on the opposite side. The surface of the river is not therefore a plane, but a complicated tvarped * Bncy. Brit., Vol. XII., p. 497. m 216 RATIO OF MEAN TO GREATEST SURFACE VELOCITY. surface, varying from point to point, and inclining alter- nately from side to side.* 116. Mean Velocity. — The mean velocity of the water in a cross-section is equal to the quotient arising from dividing the discharge per see nd by the area of the trans- verse section. When the discharge per second is not known, the mean velocity may be determined by measuring the velocities in all parts of the transverse section, and taking a mean of the results. If the transverse section is irregular in form, the only accurate manner of determining the mean velocity is to divide this section into partial areas so small that the velocity throughout each may be considered invariable. The discharge is then equal to the sum of the products of these partial areas by their velocities. Let «!, ttg, «3, etc., be the small partial areas into which the transverse section is divided, and Vj, Vg, Vg, etc., the velociti'JS in these small areas. Then the whole area is a =z flj -f ftg 4- «j -I- etc., and the whole discharge is av = a^v^ + flfgVg -t- fifjVj ■\- etc.; therefore the mean velocity is _ - 0^ + 10.33 ^^' The ratio of the mean velocity to the surface velocity in one longitudinal section is better ascertained than the ratio of the greatest surface velocity to the mean velocity of the whole cross-section. Let the river be divided into a num- ber of compartments by equidistant longitudinal planes, and the surface velocity be observed in each compartment ; then from this the mean velocity in each cojmpartment and the Wk 218 PROCESSES FOR GAUGING STREAMS. discharge can be computed. The sum of the partial dis- charges will be the total discharge of the stream. The fol- lowing formula* is convenient for determining the ratio of the surface velocity to the mean velocity in the same verti- cal. Let V be the mean and V the surface velocity in any given vertical longitudinal section, the depth of which is h. i!_ - 1 + 0.1 478 V^ ^ " 1 + 0.22ieVh' (3) ScH. — In the gaugings of the Mississippi, it was found that the mid-depth velocity diflfered by only a very small quantity from the mean velocity in the vertical section, and it was uninfluenced by wind. If therefore a series of mid- depth velocities are determined, they may be taken to be the mean velocities of the compartments in which they occur, and no formula of reduction is necessary. 118. Processes for Gauging Streams. — The dis- charge of large creeks, canals, and rivers, can be measured only by means of hydrometers, which are instrument! for indicating the velocity. The simplest of these instruments are surf ace floats ; these are convenient for determining the surface velocities of a stream, tliough their use is difficult near the banks. Any floating body can be used for this purpose ; but it is safer to employ bodies of medium size, and of but little less specific gravity than the water itself. Very large bodies do not easily assume the velocity of the water, and very small bodies, especially when they project much above the surface of the water, are easily disturbed in their motion by accidental circumstances, such as wind, etc. The floats may be small balls of wood, of wax, or of hol- low metal, so loaded as to float nearly flush with the water surface. To make them visible, they may have a vertical painted stem. In experiments on the Seine, cork balls 1| ♦ Given by Eyner in Erbkam's Zeitech. i*. for 1876. • PROCESSES FOR GAUGING STREAMS. 219 inches diameter were used, loaded to float flush with the water surface, aud provided with a stem. Bits of solid wood, and bottles filled with water until nearly submerged, have often been used for sui*face floats. Boileau proposes balls of soft wax, on account of their adhesive properties. In Captain Cunningham's observations, the floats were thin circular disks of English deal, 3 inches diameter and \ inch thick. For observations near the banks, floats 1 inch diam- eter and I inch thick were used. To render them visible, a tuft of cotton wool was used, loosely fixed in a hole at the centre. The velocity is obtained by allowing the float to be car- ried down, and noting the time of passage over a measured length of the stream. If t is the time in which the float passes over a length I, which has been previously measured, and staked off on the shore, then the velocity v \s v = ^' To mark out distinctly the length of stream over which the floats pass, two ropes may be stretched across the stream at a distance apart, which varies usually from 50 to 250 feet, according to the size and rapidity of the river. To mark the precise position at which the floats cross the ropes, Capt. Cunninghu'Q, in his experiments, used short white rope pendants, hanging so as nearly to touch the water. In this case the streams were 80 to 180 feet wide. In Avider streams the use of ropes to mark the length of run is impossible ; in such cases, recourse must be liad to some such method as the following : Let AB be the measured length = I, on one side of the river. Put two rods C and D, by means of a suitable instrument, in such a position upon the other side of the river tliat the lines CA and 1)B shall be perpendicular to AB. Then the observer, placed behind A, notes by his watch the instant the float E, which has been placed in the water some distance above, arrives at the line 220 PROCESSES FOK QAUGINQ STREAMS. AC, and then, passing down to B, he observes the instant that the float arrives at the line BD. By subtracting the time of the first observvation from that of the second, he obtains the time t in which the space I is described. For measuring the velocity below the surface, double floats * are used. They are of various kinds, usually con- sisting of small surface floats, supporting by cords larger submerged bodies. Suppose two equal and similar floats, connected by a string, wire, or thin wire chain. Let one float be a little heavier, and the othel* a little lighter an water, so that only a small portion of the latter will project above the surface of the water. We first determine by a single float the surface velocity Vg ; we tlien determine the velocity of the connected floats, whicli will be the mean of the surface velocity and the velocity at the depth at which the heavier float swims. If v^ is the velocity at the depth to which the lower float sinks, we have, calling v the mean velocity, V = 2 va = 2v — Vg. (1) By connecting the floats successively by longer and longer pieces of wire, we obtain in this way the velocities at greater and greater depths. To obtain the mean velocity in Ji perpendicular, a floating staff or rod is often employed. This consists of a cylindrical rod, loaded at the lower end so as to float nearly vertical in water. A wooden rod, with a metal cap at the bottom in which shot can be placed, so as to prevent more than the head from projecting above the water surface, answers well, and sometimes the wooden rod is made of short pieces which can be screwed together so as to suit streams of different depths. A tuft of cotton wool at the top serves to make * First used by da Vinci. he ■' MOST ECONOMICAL FORM OF TRANSVERSE SECTION. 221 the float more easily visible. Such a rod, so adjusted in length that it sinks nearly to the bed of the stream, gives directly the mean velocity of the whole vertical section in v^rhich it floats. (For a complete description of gauging streams, see " Report on the Mississippi.") 119. Most Economical Form of Transverse Sec- tion. — The best form of the transverse section must be that which presents the least resistance to a given quantity of water flowing through the channel. From Art. 103, the resistance of the bed of the stream, in consequence of the adhesion and friction, varies directly us the surface of con- tact, and consequently as the wetted perimeter p (Art. 113), and inversely as the area of the transverse section, i. e., the P resistance of the bed of the stream varies as -• In order, a tiierefore, to have the least resistance from friction, the form of the section must be that which has the least perim- eter for a given area, i. e., the wetted perimeter/; must be a minimum for a given area (u or the area must be a maximum for a given wetted perimeter. Now, among all figures of the same number of sides, the regular one, and among all the regular ones, the one with the greatest number of sides has the smallest perimeter for a given area. Hence, for closed pipes, the resistance of friction is the smallest when the transverse section is a circle ; but in open channels, the upper surface, being free, or in contact with the air I f >- — . /- alone, must not be in- cluded in the perimeter. A horizontal line DC, passing throu/nh the centre of the square AF, divide.s the area and perimeter into two e(jual parts, and what has been said of the square is true of these halves ; hence, of all rectangular forms of transverse sections, the half square 232 TRAPEZOIDAL SECTION OF LEAST RESISTANCE. ABCD is the one which causes the least resistance of fric- tion, and therefore is tiie best for open channels. Also, of all trapezoidal sections, the semi-hexagon ABCD is the one which causes the least resistance of friction ; and so on to the other cases. But the semicircle will present less resist- ance of friction than the semi-hexagon, and this latter less than the semi-square. The half decagon oflFers still less resistance than the half hexagon or the half square. The circular and sfjuarc sections are used only for troughs made of iron, stone, or wood. Tlie trapezoid is employed in canals, which are dug out or walled up. It is very rare that other forms are used, owing to the difficulty of constructing them. 120. Trapezoidal Section of a Canal of Least Resistance, when the Slope of the Sides is Given. — Let ABCD be the section. Put X = AB, the width of the bottom, y =z BE, the depth, and 6 = BCE, the angle of the slope, which is to be considered as a given quantity, dependent upon the nature of the ground in which the canal is excavated, and a = the given area of the section ABCD. Then the wetted perimeter of the section is _D F Fig. 63 ;> = AB + 2BC = X -{■ 2y cosec 0. The area of the section is a ■= xy •{- y^ cot 6 ; a — w2 cot 1- — ^ (1) y which in (1) gives __a — y^ cot B p = -f 2y cosec 0. (2) (?) i EXAMPLE. 223 To find the value of y which makes this a minimum, we must equate to zero its derivative with respect to y, which gives yi = a sin 2 — cosO' sin Va s cos 6 (4) (5) Hence, for a given angle of slope 0, and for a given area «, the trapezoidal section of least resistance is determined by (2) and (5). . Consequently, the width CD of the top IS CD = X -\- 2y cot d a _- _ -i. « cot 6 ; (6) and the value of ^, from (3), is p 1 2 — cos a y 2 a sin y = - [from (4)]. y EXAMPLE- (7) What dimensions shonld be given to the transverse sec- tion of a canal, w! ,n the angle of slope of its banks ,s to be 40°, and when it is to carry 75 cubic feet of water with a mean velocity of 3 feet ? Here we have ^ — -1^- = 25 square feet; and hence, from (5), we have the depth 224 UNIFORM MOTION. From (2), we have the width at the bottom, 25 X = — 3.609 cotan 40' 3.609 = 6.927 — 4.301 = 2. M feet. The width of the top, f 'om (6), is CD = 2.626 + 7.218 cot 40° = 11.228 feet The wetted perimeter is ^ ^ sm 7 218 = 2.626 + . ,\,o = 13.855 feet : sin 40 and the ratio which determines the resistance of friction is £ ^2 y a 0.5542. Rem. — In a transverse section in the shape of the half of a regular hexagon, where d = 60°, ./; = 4.39, y = 3.80, width CD = 8.78, and p = 13.16 feet, we have, for the resistance of friction. a 13.16 25 = 0.526, which is less than that found for the above trapezoid. 121. Uniform Motion.- -When water flows in an open channel, the velocity continues to increase so long as the accelerating force exceeds the resisting force of friction ; but when these forces are equal to each other, the velocity of the stream becomes uniform. When the velocity is uni- form, the entire head h is employed in overcoming the fric- tion upon the bed, Therefore, the height of the column of 1 18 of he ^UNIFORM MOTION. %%h water due to the resistance of friction must be equal to the 7alL The height due to the resistance of friction increases with ^ , ^vith the length /, and with the square of the ve- locity I (Art. 102). Hence, from (1) of Art. 103, we have Ip v^ (0 in which /is an empirical number, which is called the coef ficient of friction. Solving (1) for 1', we have /2gha (2) Accordins to Eytelwein's reduction of the ninety^ne ob- A = 0.007565 |.^- (3) If we put g = 33.2 feet, (2) and (3) become /ah A = 0.00011747 1 v\ (6) For the number of cubic feet of water flowing through the channel per second, we have (6) /ah Q=:av = 92.26ay j^- Cor.— For pipes, we have ?P_ a 22C COKPFtrtESTS OF FlfWTlON. which in (3) gives A = 0.0302(} ,.-» which agrees witL (5) of Art. 104 and (1) of Art 105. EXAMPLE. How much fall must a canal, whose length is 2600 feet, whose lower width is 3 feet, whose upper width is 7 feet, and whose depth is 3 feet, have in order to ctirry 40 cubic feet of water per second ? Here we have jt) = 3 + 2\/22 + '^^ = 10.211, and _ 40 _ 8 *' ~ 15 ~ 3' Substituting in (5), we have 2600 X 10.211 /8\8 n = 0.00011747 15 § 0.30542 2 X 10.211 x 64 15x9 = 1.48 feet. 122. Coefficients of Friction.— The coefficient of friction / varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity, as in the case of pipes, increasing slightly when the velocity diminishes, and decreasing when the velocity increases. A common mean value assumed for / is 0.007565, which we used in the last Art., though it has quite a range of values. Weisbach, from 205 experiments, obtained for / the follow- ing values at different velocities : EXAMPLE. m (7) V = 0.3 0.01215 0.4 0.01097 0.5 0.01025 0.6 0.00978 0.7 0.00944 V = f = 0.8 0.00918 3 0.00788 0.9 0.00899 1.0 0.00883 0.00836 2 0.00812 15 0.00750 V = 5 0.00769 7 0.00761 10 0.00755 In using this table for the value of/ when v is not known, wc must proceed by approximation. Determine v approxi- mately from (4) of Art. 121. Then from this value of v find /by means of the table, and substitute the value of / so found in (2), and determine a new value of v. EXAMPLE. What must be the fall of a canal 1500 feet long, whose lower width is 2 feet, upper width 8 feet, and depth 4 feet, when it is required to convey 70 cubic feet of water \)er second? Here we have p = 2 -\- 2\/l6 + 9 = 12, « = 5 X 4 = 20, v = U = d.5; hence, from the table, /= 0.00784. Substituting in (1) of Art. 121, we have 1500 X 12 S^^ h = 0.00784 86.436 64.4 20 " = 1.34 feet. ^9 228 VARIABLE MOTTOX. 123. Variable Motion.— In every stream in which the discharge is constant for a given time, the velocity at differ- ent places deiKjnds on the slope of the bed. In general, the velocity will be greater us the slope of the bed is greater; and, as the velocity varies inversely as ^he transverse section of the stream, the section will be leaitt wliere the velocity and slope are greatest. In a stream in wiiicli the velocity is variable, the work dne to the fall of the stream for a given distance is equal to the work destroyed by friction together with the kinetic energy corresponding to the change of velocity, /. e., the whole fall is the sum of that expended in overcoming friction, and of that exj)ended in increasing the velocity, when the velocity increases, or if the velocity de- creases, the head is the difference of these quantities.* The resistance of friction upon a small portion of the length of the stream may be regarded as constant and meas- ured by a head of water (1) Fig. 64 Let ABCD represent a longitudi- nal section of a short portion of a stream, AB the surface of the stream, and AE and HG two horizontal lines. Let I = the length of AB in feet; h = BE, the fall from A to B; Vq = the velocity of the stream at the upper section AD ; and Vi = the velocity at the lower section BC. Now the velocity of any particle B, at the surface of the stream, is due to the height h, together with the velocity at A ; hence we have, for its velocity i\, V ^9 i- = A + "^^ '^9 i:^) * In long rivers, with slopes not greater than 3 feet per mile, the velocity head is aeually insignificant compared with the friction head. (See Fanning^s Water- Supply Engineering, p. 903.) VMirAnijE MnTioK. m (1) (9) Any particle (I, bciioutl) the surface oi the water, is pressed forward by the lieud AH = EG, and pressed biu3k- ward by the head BU ; hence tlic lieud which produces mo- tion is EG — BG = EB, or h, as before ; and therefore (2) is true for any particle. Solving (2) for h, and adding the resistance of friction, as given in (1), we have h = V a _ 2g '^•' a 2g* (3) in which p, a, and v denote tlie mean values of the wetted perimeter, the transverse section, and the velocity, respect- ively. If Oq and a, denote the areas of the upper and lower transverse sections, respectively, and Q the quantity of water which flows through any section in a unit of time, we have a = _ «o 4- (ii 2 and Q = a^fV^ = ajv,. From (5), we have 2g \a,^ a^V ^ (4) (6) (6) Now if the water flowed with the velocity v^, we would have the head due to the resistance of friction, from (1), -'a 2g' (7) and if it flowed with the velocity Vj, we would have the head due to the resistance of friction _ lpv_l ■'a 2g (8) But the former expression is less, and the latter is greater than the true head due to the resistance of friction; hence, ■' I i 230 VARIABLE! ifOTlON. the mean of these results will give the friction head approx- imately. Therefore, taking the mean of (7) und (8), and substituting for a its value from (4), we have, for the re- sistance of friction, (9) [from (5)]. Substituting (6) and (9) in (3), we liave, for the whole head, Solving (10) for Q, we have Q = V2gh V «i^ «o' + /:; ip «o + tf, Uo^ "^ «,V (11) In a prismoidal channel it will be a sufficiently close ap- proximation to the truth to assume that tiie surface line of the water is straight, and then from this assumption to com- pute the transverse sections and their perimeters. When we have these, with the quantity of water carried and the length of a portion of the river or canal, we may determine the corresponding fall h by (10) ; and when we have the length, fall, and cross-section, we may determine the quan- tity Q by means of (11). Where greater accuracy is re- quired, we should calculate h or Q for several small portions of the stream, and then take the arithmetic mean of the results. If only the total fall is known, this value should be substituted for h in (11), and instead of — s ^ we a. «o^ EXAMPLE. 231 (9) should use — ^ ^, whore an denotes the area of the low- ttf? a, est transverse section, and instead of i_ (J /: r,\r-^ + ajl' the sum of all the similar values for the different portions of the stream should be used. (See Weisbach's Meclis., p. 9G9; also Tate's Mech. Phil, p. 305.) J EXAMPLE. A stream falls 9.6 inches in 300 feet, the mean value of its wetted perimeter is 40 feet, the area of its upper trans- verse section is 70 square feet, and that of its lower is HO square feet. Find the difc,charge of this stream. From (11), we have Q = 8.025\/0.8 V'^-^^+ 0.007565?!^ 300 X 40 / 1 30 \602 ^ 702/ 7.178 = 354| cubic feet. VO. 0000731 + 0.0003365 The mean velocity is %Q 709 - .- - . -— = -^^ = 5.45 feet ; a^ -\- a^ 130 hence (Art. 122), a more accurate value of /is 0.00768, and therefore we have Q = 7.178 V0.0000r31 + 0.0003416 = 352.5. If the same stream has at another place a fall of 11 inches in 450 feet, and if the area of its upper transverse section is 11 232 BOTTOM VELOCITY AT WHICH SCOUR COMMENCES. 50, and that of its lower GO square feet, the mean value of its wetted perimeter being 30 feet, we have 8.025 a/0.916 /I l" . AnA^«o450x3G/l , 1 \ / ~ 0.9107 - ^-^^^V -l):000l222"+0: 000?549 = 305^ cubic feet. The mean of these values is _ 352.5 + 305.5 = 329 cubic feet. ScH. — The following is Chezy's formula, with three dif- ferent coefficients, varying from 09 "for small streams under 2000 cubic feet per minute," to 96 ''for large rivers such as the Clyde or the Tay." V = 69 (r sin dp. For small streams, v = 93 (r sin 0)'). Eytelwein's coefficient, v = 96 (/• sin 0)K For large streams. 124. Bottom Velocity at which Scour Com- mences. — A river channel is said to have a fixed regimen, when it changes little in draft or form in a series of years. In some rivers, the deepest part of the channel changes its position perpetually, and is seldom found in the same place two successive vears. The sinuousness of the river also changes by the erosion of the banks, so that in time the position of the river is completely altered, in other rivers, the change from year to year is very small, but probably the regimen is never perfectly fixed except where the rivers flo'.v over a rocky bed. If a river had a constant discharge, it 'ES. TRANSPORTING POWER OF WATER. 233 le of would gradually moaify its bed till a permanent regimen was established. But as the volume discharged is constant- ly changing, and therefore the velocity, silt is deposited when the velocity decreases, and scour goes on when the velocity increases in the same place. It has been found by experiment * that a stream moving with a velocity of 3 inches per second will carry a\ong fine day and so/i earth ; moving 6 inches per second, will carry loam J 1 foot per second, will carry sand; "Z feet per f-econd, gravel j 3^- feet, pebbles an inch in diameter ; 4 feet, broken stone, flint ; 5 feet, chalk, soft shale; 6 feet, rock in beds; 10 feet, hard rock. 125. Transporting Power of Water.— The specific gravity of rocks varies from 2.25 to 2.64; when immersed in water, therefore, they lose nearly half tlieir weight. This fact greatly increases the transporting power of water. The pressure of a current of water agjiinst any surface varies as the square of the velocity and as the arcft of the surface f (Art. 97). Put in similar figures, surfaces vary as the squares of the diameters; hence, tiie jiressure of (lie current varios as the square of the velocity and as tiie square of the diameter, i. e., the ]jressu»'o of the current against a surface varies as the square of its velocity multiplied by the square of the diameter of the surface. Calling P tiie pressure which the current exerts against a r«)ck, v its velocity, and d the diameter of the surface of the rock, we have a V xd\ (1) Now the resistance to be overcome, or the weight of the rock, varies as the cube of the riitimeter: /, ,:., calling W the weight of the rock, we have IT' ex. d^. (2) * Experiments by Dubuat. See Ency. Brit., Vol. XII., p. 608. + SappoHinf; that the area of the cro8f>>ti, the work to be done = Pi^'i log ^> Fi (8) which is the iroi'Jc of com pressing a given mass of air from a lower pressure p^ to a higher pressure p^. ScH. — The expressions in (2) and (3) for the work done during the expansion and compression of air, are correct only when the temperature of the air remains constant while the change of volume or density is taking place ; but the temi)erature of the air remains constant only when the change of volume takes place so slowly that the heat in the confined air has sufficient time to communicate any excess io the walls of the vessel and to the exterior air. If the change of density occurs so quickly that it is accompanied by a change of temperature, when the air is expanded the ♦ Hyp. log. 244 VELOCITY OF EFFLUX OF AIR. temperature is lowered, and when the air is compressed the temperature is increased. Under these circumstances the pressure cannot chan{;^e according to Art. 48, and other formula} have to be produced. (See Woisbach^s Mechs., p. 936 ; also Ency. Brit., Vol. XII., p. 480.) 129. Velocity of Efflux of Air According to Mariotte's Law. — Let the air be discharged from an orifice with the velocity v feet per second; let w = the weight of a cubic foot of air, and v^ = the volume of air discharged per second; then the work performed by the volume of air Vj in passing from the pressure p^ to the pressure jOg, is, by (2) of Art. 128^, Px^x log ^, and this must be equal to the work stored in tlie air during the efflux, which is Therefore, we have = Px^x log ^, ^g Pi ••• ^ = \/ .Pt w 2^ — log^. Pa Pi (1) A cubic foot of air, at the temperature 0° of the centi- grade thermometer, and at a pressure corresponding to the height of 29.92 inches of the barometer, weighs about 0.08076 lbs.* Therefore for any temperature /, we have for the weight of a cubic fc^^t of air, from (2) of Art. 54, since for the same volume and temperature the weight varies as the density, 0.08076 w = 1 + at (2) * Determined by Regnaolt. See Weisbach'a Mechs., p. 796. Bd the ;es the other hs., p. (1) VELOCITY OF EFFLUX OF AIR. 245 I If the pressure differs from the mean pressure, or if the height of the barometer is not 29.92 inches, but b, (2) be- comes 0.08076 b w = 1 -\- at 29.92 0.C02699 b If we express the elastic force or pressure of the air by the pressure jt) upon each scjuare inch, then we have b _ p 29^92 ~ 147' which in (3) gives a 005494 » w = ^ . 1 4- «< p _ (1 + a t ) 144 '' w~~ 0.005494 ' where p is the pressure on each square foot. Substituting this value in (1), Ave have (3«) (4) (6) V = 161.9 \J'Zg (1 + «0 log^. (6) If b is the height of the barometer and h that of the manometer (Art. 46), we have which in (6) gives • V = 1299 Px _ b-\-h ^(l-f«/)logf->^), (6a) (7) where v is the velocity in feet, b the height of the barometer in the exterior air, // the height of the manometer for the air inside the vessel, / the temperature of the latter in degrees centigrade, and « = 0.003665, the coefficient of expansion of air (Art. 53). 246 VELOCITY OF EFFLUX OF AIR. ! II Pi i \ Cor. 1. — If the pressures p^ and p^ are nearly equal to each other, we can put h\ h h^ •»^m='"«(^+t)=f-4. which in (7) gives V = 1299 ^(l+ecO(l-^j h 2b/ b \ (8) h When T is very small (8) becomes V = 1299 a/ (1 + cct) |. (9) CoK. 2. — Taking g = 32, we have from (1) (10) When the pressures differ but little from each other, we may obtain approximate formuliu as follows : From (10), we have By development we have Neglecting all powers of P9 -Pa above the first, and sub- stihiting in (11), we have -Ih (12) EFFLUX OF MOVING AIR, 247 ^ual to : (8) (9) Neglecting all powers of — — ^ above the second, we have V\ If h be the height of a homogeneous fluid, of the same density as the air, which is necessary to produce the pressure Pi —Pif tlien jOj —Ps = ^^'/^ which in (12) gives V = SVh. (14) It will be observed that (8), (9), (12), (13), (14) are true only when the pressures jw, aud jOg are nearly equal to each other. . (10) ler, we 1 (10), (11) •Xj» sub- (12) 130. Efflux of Moving Air,— To find the velocity of efflux ivhen the pressure of the air is given in the pipe through which it fiyows, "^he formula) for efflux found in Art. 129, are based upon the supposition that tiio pressure jo, or the height //, of the manometer is measured at a place where the air is at rest, or moving very slowly. If the pressure be measured at a point where the air is in motion, in determining the velocity of efflux, >ve must take into account the kinetic energy of the moving air. Let p^ be the pressure of the air in the pipe A, as indi- cated by the manometer M, and v^ the velocity of the air passing the orifice of the manometer; j^g the pressure of the air at efflux, and v^ its velocity; a^ the area of the section of the pipe A, and «g the area of the orifice ; w the weight of a ('ul)ic foot of the air, and O the volume dis- charged per second. Then the work stored in the air while passing from the pipe A to the orifice Fig. 68 -ei Jk^ 248 EFFLUX OF MOVING AIR. = ||'(V-V); and this must equal the work doue by the expansion of the air from jt>, to p^. Therefore, from {2) of Art. 128, we have ^(v-f,») = ?ieiog|| (1) The volume of air passing through the pipe per second is a^v^, and that which passes the orifice is ajVjj ; hence we have (Art. 48) which in (1) and reducing, gives .*. Vj = a^pi Vg, ^9Pt log ~ Pz (2) which is the velocity of efflux. Cor. — Sub.^tituting for the numerator its value as given in (8) of Art. 129, we have V, = 1299 (i + «^)(i-4)* 1 _ {^Pl^ or approximately, when jOj is not much greater than jt>g, (3) v. = 1299 / (1 + «0i (4) COmF^WtKXT OP EFFLUX. 249 of the 128, we (1) Jcond is ence we '8> (2) 8 given «> (4) 131. Coefficient of Efflux.— When air issues from an orifice, the section of the current undergoes a contraction similar to that observed in the efflux of water (Art. 91). If the orifice of efflux is in a thin plate, the stream of air has a smaller cross-section than the orifice, and the practical discharge is less than the theoretical. Denoting the coefficu.'U of contraction by «, we have, as in the case of water (Art. 92), « = the ratio of the cross- section of the stream of air to that of the orifice. Denoting the coefficient of velocity by , we have, as in Art. 93, = -^ , where v, is the actual and v the theoret- ical velocity of discharge. Deroting the coefficient of efflux by //, we have, as in Art. 94, \i Y/(l + at) log (*-±-i^). (4) From (4) of Art. 130, we have Q =. 1809,.i / *. (5) COEFFICIENT OF FRICTION OF AIR. 251 |o 0.788 »o area per sec • Then (1) I; (3) (3) G the (4) 133. Coefficient of Friction of Air.— Wlien air flows through a long pipe, it has, like water, a residanre of fiir- iion to overcome, due to tlie surface of the pipe ; and tliis resistance, which is found to consume by far the greater part of the work expended, can be measured by the height of a column of air, which is determined by the expression. (1) (5) in which, as in the case of water (Art. 103), / denotes the length, d the diameter of the pipe, v the velocity of the air, and / the coefficient of resistance of friction, to be deter- mined by experiment. The Avork expended in friction gen- erates heat, the most of which must be developed in the air and given back to it. Some heat may be transmitted through the sides of the pipe to surrounding materials, but, in all the experiments that have thus far been made, the amount so conducted away appears to be very small ; .and if no heat is transmitted, the air in the pipe must remain sen- sibly at the same temperature during expansion ; that is, the heat generated by friction exactly neutralizes the cool- ing due to the work done. A discussion by Prof. Unwin * of the experiments by Messrs. Culley and Sabine on the rate of transmission of light carriers through pneu- matic tubes, in which there is a steady flow of air not sensibly affected by any resistances other than the surface friction, furnished the value / = 0.028. The pipes were of lead, slightly moist, %\ inches (0.187 ft.) in diameter, and in lengths of 2000 to nearly 6000 feet. Qirard's experir.ents upon the motion of air in pipes gave a mean coefficient of resistance, / = 0.0256 ; those of D'Aubuisson gave as a mean, / = 0.0238 ; while those of Buff gave the mean value of / = 0.0375. According to the experiments of Weisbach, it is only when veloci- ties are about 80 feet that the coefficient of resistance can be put = 0.024, and it diminishes as the velocity of the air in the pipe increases. * See Ency. Brit., Vol. Zn., p. 252 MOTION OF AIR IN LOSa PIPES. * «! He found that the coefficient of friction, when the velocity was giv -.1 in feet, could ^)e expressed approximately by the foUcv.n^ fonaulu, / = 0.217 r^) Tb:* resistant ^ caused by elbows and bends is to lie treated in the same way as iu m ^ case of water (Arts. 110, 111). 134. Motion of Air in Long Pipes.— By the aid of tl»e coefficient of friction of a jupe, we can calculate the velocity of efflux and the discharge for a given length and diamete) of the pipe. Let V = the velocity of discharge = ¥ • Vi = the velocity of the air in the pipe. d = the diameter of the orifice, whose area there- fore is k = ^nd^. rf, = the diameter of the pipe. j3p = the coefficient of resistance at the entrance to the pipe. / = the coefficient of resistance due to the friction of the pipe. f3^ z= the coefficient of resistance at the orifice. Pi = the pressure of the air when it is discharged. w = the weight of a cubic foot of air. h = the height of the manometer in the reservoir. b = the height of the barometer. I = the length of the pipe. Then the height due to the resistance at the entrance to the pipe, - 3 ^ - fl d*_^ d,' 2g' The height due to the resistance of friction in the pijx* I V, — J .1 o« -^ J I d^ ^ d^2g ''d^d,*2g ras giv •:. friiiula, M in tlie aid of |ate the :th and there- mce to riction •ged. •voir. ice to i[)e MOTION OF AIR IN LONG PIPES, Thf height due to tlie resistance at the orifice 853 ^/^i.: 2^ The height due to the velocity _ i? Therefore, the total height = ^-^/i),f.---.J4(l)- (1) * Also, tlie total height, from (1) and (6a) of Art. 129, = ^ log (l + ^) = -^ -^, approximately. (2) Therefore, equating (1) and (3), and solving for Q, we have Q = Tc ^9 10 h which, from (9) of Art. 129, = 1299!^^ (1 + «0 I (^o+/i)(|)Vl+/^. , (3) where (i^ = —.- 1, and fi^ = — „ - 1 (Art. 96). ' ~ N' ScH. — In Paris, Berlin, London, and other cities, it has been found cheaper to transmit messages in pneumatic I S54 LAW OF THE EX PANS/OX OF STFAif. iSU tubes tiian to tele^apli by oloofricity. Tl»o tiil)cs arc laid under ground, witli easy curves; I ho nu'Ssagi'S mv uuult' into a roll and plucod in a light Iblt carritT, the resistantT of which in the tubes in London is only | oz. A current of air, forced into the tube or drawn through it, propels the carrier. In most systems the current of air is steady and continuous, and the carriers are introduced or removed without materially altering the flow of air. 135. The Law of the Expansion of Steam.—When Bteam is produced in a close vessel, as in the boiler of a steam engine, the density of the steam increases with the temjierature ; but so long as the temperature remains the same, the quantity of steam that can be raised from the water is limited, suid the steam is generated at its maximum density and pressure for the temperature, whatever tiiis may be; if the tcmporaturo falls, a portion ojC the steam resumes the liquid form, and the density of the steam is diminished. When the steam is in its condition of maxi- mum density, it is said to be saturated, being incajyable of vaporizing or absorbing more water into its sul)stance, or increasing its i)ressure, so long as the temperature re- mains the same. Also, on the contrary, steam will not be generated with less than the maximum (puintity of water which it is capable of ai)pro}>riating from the liquid out of which it ascends. Anyehansfe in either one of the three elements of pressure, density, or temperature of steam is necessarily accompanied by a change of the other two. The same density is invariably accom]>anied by the same pressure and te?ni)erature. If the volume of steam over water be inc^rcascd, while the temperature remains constant, then, as long as there is liquid in excess to supply fresh vapor to occupy the increased space, the density will not be diminished, but will remain constant with the pressure. If the source of heat be re- moved, when all the liquid is evaporated, the pressure and LA W OF THE EXPANSION OF STEAM. 255 laiid lit of the and loved density will diminish, wiien the volume is increased, us in j)t'rnianent guses; uiid if tlie volume Ix^ aguin diminished, llie i)re8sure .nd density will increase, until they return to the maximum due to the temperature ; and the effect of any further diminution of volume, or attempt to further in- crease the density at the same temperature, is simply accompanied by the precipitation of a portion of the vapor to the li(piid state, the density remaining the same. On the contra.^, if the application of heat be continued when all the li(|uid is evaporated, the state of saturation ceases, and the temperature and pressure are increased, while the density remains the same ; the steam is said to be superheated, or surcharged with heat, and it becomes more perfectly gaseous. While in this condition, if it were to be replaced in contact with water of the original temperature, it would evaporate a part of the water, transferring to it the surcharge of heat, and would resume its normal state of saturation. If the space for steam over the water remain unaltered, then, if tho temperature is raised by the addition of heat, the density of the vapor is increased by fresh vaporization, and the elastic force is consequently increased in a much more rapid ratio than it would be in a permanent gas by the same change of temperature. Conversely, if the tempera- ture be lowered, a part of the vapor is condensed, the den- sity is diminished, and the elastic force reduced more rapidly than in a permanent gas. The density of saturated steam is about | of that of atmospheric air, when they are both under the same pressure and at the same temperature. It bus been determined experimentally that whatever may lie the pressure at which jteam is formed, the quantity of fuel necessary to evaporate a given volume of water is always the same ; also the relation between the temperature and pressure of saturated steam has been determined experi- mentally, and from this tables have been formed giving the relation between the pressure and volume of stean) raised i 25G * WORK OF EXPANSION OF STEAM. from tt cubic foot of water.* Since the volume is ulwuys a function of the pressure, we nuiy write F=:/(n (1) I - A B Fig. 69 136. Work ()f KxpaiiMion of Steam. — Tict V be the volume of steam from a cubic foot of water at I lie pressure P, where P is the pressure on a 8(iuare foot, anil Fq the work performed by Q cubic feet of water, in the form of steam, between the pressures P and Pj ; let ABCD be a vertical section of the space in winch the steam expands, AHPQ the volume V of the steam at the pressure P, ABPjQ, the volume Fj of the steam at pressure /\, A the area of the section at PQ in feet, and dv the dis- tance between the two consecutive sectiotis PQ, and MN. Then, for the element of work performed by one cubic foot of water in the form of steam at the pressure P, we have dlJ^ = AVdv = PdV, since Adv := dV. Integrating between the limits P and Pi, we have [from (1) of Art. 135]. Therefore, for the work done by Q cubic feet of water in expanding from P to P^, we .'uive V Uq=QfJpdf{P), (2) which can be integrated when the function f{P) is known. Sen. — Since from (2) the quantity of work done is en- tirely independent of the form of the vessel ABCD, it - . . • See Eucy. Brit., Art, Steqm. WORK OF STEAM AT EFFLUX. 267 (1) Ic the IsHiire Miu I'll I of follows that the work of steam between any given pressures P and P, is always the same, whatever may be the uuturc of the space through which tlie steam expands, and that it increases with the pressure /* at which the steam is gener- ated ; since therefore the quantity of fuel necessary to evaporate a given volume of water is always independent of the pressure at which the steam is formed (Art. 135), it fol- lows that it is most economical to enploy steam of as high a temperature as possible. 137. Work of Steam at Kfflnx.— Let w be the weight of Q cubic feet of water evaporated per second, V the volu.ne of steam from a cubic foot of water at the press- ure P, which is the pressure of the steam at tiie point of efflux, k the section of the orifice in feet from which the steam is discharged, and v the velocity per second. Then calling Uq the work stored in the steam at efflux, we have Since V=:f{P), and QV = kv, we have w kv=Q/(P) = ^^^f(P); (1) w which in (1) gives •■• " = (i2.5X-^(^>' 1V^ f^« = r8i^5;iS,[/'^'J'- (2) (3) Hence, the work of stcanidiHchai'^m^ Itself from an orifice varies as the en he of the water ivaporated. Cou. 1. — The work stored in the steam at efflux is due to the work of expansion between the pressures P and P, ; '! ; ■rf 258 WORK OF STEAM AT EFFLUX. therefore, rrom (2) of Art. 130 and (1) of the present Art., we have ••• ^->'{i,.Q''V('')^' (S) irhich glres the irlocitij of vfjlwx, ft heln^ the cncjficicnt of r/ff,ii,\' {Art. V3l). Cor. 2. — From (1) of Art. 130, we have («) Let r, tuul Fg he tlie vohimes of nteam per seeoiul from a eiihie foot of water at I\ and P^ pressures respectively; then (Art. 130), lit V ^2 = Q'ry (U wliicli in (0) gives d) and we see again, as in (3), tlisit the ivorh of stcain varies (ts fhc cube of the initcr evaporated. ►Solving (7) for (^> ' = /•'-*, we have tti iH< — («) u'hi eh gives the theoreileal veloeifif of ejflux. WORK OF STEAM IN THE EXPANSIVE LNGINE. «i60 Art., (4) \cient («) (7) 138. Work of Steam in the E? pausive Engine.-- Lot K be the area of tliu piston in square feet, h^ the length of tikc stroke, including the clearance,'" h the point of the cylinder at which the steam is cut off, Q the number of cubic feet of water evaporated per minute, P and 7*, the l)ressure8 of the steam at the beginning and end of the stroke res|)ectivoly, N the number of strokes performed by the piston |)er minute, V the volume of steam from a cubic foot of water at P pressure, and Vz=f{P), as before. Then, for the volume of steam discharged per minute at P pressure, we have Qf(P) = NKh. (1) Similarly, and fil') - A (3) The work performed upon tiie jjiston before the steam is cut off is iVA7' {h — c) ; adding this to (2) of Art. 130, we have Uq = QfpdfiP) + NKP{h - c) P, ■ oW' ^ifin + -r-PfiP)] [from (I)], which is the toted work jwrfonned htj the steam ftcr miiNife. Cor. — Ijct L be the useful load in lbs. u])on each square foot of llu! piston, F the friction in li/s. per s<|uare foot of tiie piston, iirisin<; from i\\v niolion of the Mnlo;id(>«l piston, /'the ciieirK-icnt of I'riclion iirisiM;^^ Iroin (he u^tTnl load, and * Tlio itearance Ih thu Hpacu iti tbe cylindur lying beiiuatb the piHtoD, at UiO low* CHt polDt of iu btrukv* ff 260 EXAMPLES. the of the Hicuiii in tb, idenscr. Then th prcRfiurc total resiHUiiice uiKiii the piston is A' [/'J- Jj (1 +/) -+- ;>), ttnd therefore tl»e work expended per n»lnute in overcoming tills resistance = NK[F^ L (1 + /) -f p] (A, - r). (4) When the mean motion of the jiiston <»f tii(> engine is uni- form, the work of the resistauee will Iw e(|ual to the work of the steam ; therefore, l)y equating (3) and (4), and re- duoing by (1), we have = [/'■+/-(! +/) + ;,)(/,, -r), (5) from whieh the value of the useful loud A is readily de- termined. EXAMPLES. 1. If a blowing maehine ehaMgcs per second 10 cubic! feet of air, at a pressure of 28 inches, into a bUist at a jiressure of 'M) inches, find the work to be don(! in each settond. II«ro;)g, from CJa) «>f Art. 121), O.-IDNM* ; .-. i'U-,. Jus. i:m\.7 foot-lbs. 2. If under the piston of a steam engim>, whost^ ar(>a is 201 K(|uaro incthes, tlutre is a ((uantity <»f steam ir> iiu;hcs high and at a pntssure of ;( ainiosphcrcs, and if this steam in expanding moves tlu; pisioii forward 25 inches, find the work of tiie expansion per s(!cond. A us. lOHOd foot-lbs. .'J. The air in a reservoir is at a Icmperalure of 120" (!., and at a pressure corresponding lo a height of the manom- eter of ft inches, whih^ Ihe biinmieler ..larks 2!>.2 ineheM. rind (I) Ihe Ibeoreliejd veloeily cd" eMlux, and (2) the iheo- reti(;al disc^harge Ihniugh an oriliee ]\ inches in liaroineter stands at ;;!0 inches and the icinperatiire of tlut air in the pipe is 20° 0. Ans. (1) .M7.()(; feet; (2) '.'.4;}H cnhic feet. G. If the Hiun of the areas of two conical tiiyen's of a blowing machine is '.] si|>e 000 feet long and G inches in dianu'tur, tho lu'iglit of liio barometer 30 Indus, the diameter of the orilice in tiie eonieally convergent end of the ]»ipe 2 inelies, and the temperature of the compressed air in the regulator JJO" C, llnd the quantity discharged. Ana. 9.051 cubic feet. 1). If the height of tlie manometer in Ex. 7 is 2.5 inches, the i)ipo GOO feet long and 5 inches in diameter, tho heigiit of tho barometer 29.5 inches, the diameter of the orilice in tiie eonieally convergent end of the pipe one inch, and tho temperature of tiie compressed air in the regulator is 10" C, lind the quantity discharged. Ans. 1.88*3 cubic feot. kf^ : % plo is iH'tor, If I ho M'lics, ilufor |c'I)OS, 30 in tho »" a. CHAPTER IV. HYDROSTATIC AND HYDRAULIC MACHINES. ll{9. Dofliiitions. — 'Vhcrc arc several 8imi)le machiiu's wlioso ad ion (U'peiuls on the itropcrtii'S of air and water ; a brief description of some of these machines will now he given, snflicient, to exhibit tho principles involved in tlieir constrnction and nse. ililherto the energy exerted by moans of a head of water has been wholly emph»yed in overcoming friotional resist- ances, and in gerioraliiig tlie vcdocity with wliicli tiic water is delivered at some given ])oiMt. In the cases which wc have now to consider, only a fraction of the head is required for these purposes; the remainder, therefore, becomes a Siuirce of energy at the point of divery by means of which useful work nuiy bo done. Hydraulic energy may exist i three forms, according as it is due to inoliou, elevation. '\- pressure. In the first two cases the energy is inherent the water itself, being a con- sccpienee of its motion <" . -sition, as in the case of any otiier heavy body. In th ihird it is due to the action of gravity or some other I'orci', sometimes on the water itself, but oftener nu other bodies : the water then only transmits the eutM'gy, and is not directly the source of it.* 140. Tlie Ilydrostati*' Bellows. — This machine pre- sents an illustrailion of tlu' juinciple of the transmissi(»n (»!" fluid pressure (Art. 8). il consists of a eyliiider ('I)KF (Fig. 70). ^vitli its sides made of lealher or other flexible material, and a pipe /iiil'' l''ading into it. If water is * Cott«ri>.'rt A|>|>. Mvcbu., |>. 4t)9, :eG4 THE SIPHON, ! i poured into the pipe till the vessel and i)ipe ure filled, a very small pressure applied ul A will raise a very great weight upon DE, the weight lifted being greater us DE is greater. Let k be the area of ahorizontalsection of the i)ipe, K that of a seetion of the cylinder, or that of DE, and p the i)ressiire applied at A. Then, from (1) of Art. 0, we have ¥ ~ K (1) Fig. 70 Rrn. — Suppose the pipe AH to be extend- ed vertically upwards, and the pressure at A to be produced by means of a column of water above it, formed by pouring in water to a considerable height, and 8upp«)sc the pipe to be very small, so thut the pressure upon the section A may be very small; liieu, as this pressure is trausmitted to every jmrtion of the surface DE that istxpuil to the seetion A, the upward force produced on DE can be as large as we j)lease. To increase the upward force, we must enlarge the surface I)K oi- increjise the height of the column of water in (lie pipe, and the only limitation to the increjiso of the force will be the want of sutticient strength in the ])ipo aiul (cylinder to resist the ifurreased pressure. By making the pijut AH of very small bore, and the height DC of the cylinder very snudl, the ((uantity of water can be made as small as we please. That is, anij qnantitif of fluidy hoivcver snitill, nun/ he ni(i(lr to supfxtrf onij weight, however grtuU. This is known us " e hydros/aiic paradox. 141. Tll<^ Siphon. — The action of ii sipjio)/ U an i>n- portant practical illustration of atmospheric pressure. It is sini])]y a bent tube of unequal braiu'hes, open at both ciuls, and is used to convey a liquid fnmi a higher to a lower level, over an intermediate point higher than either. w a I THE sii'uoy. 2G5 Let A and B be two vessels containing water, U being on tiie lower level, and AOB a bent tube. Suppose this lube to be filled with water from the vessel A, and to have its extremities immersed in the water in the two vessels. The water will then flow from the vessel A to B, as long as the level B is below A, and the end of the shorter branch of the siphon is b«low the surface of the water in the vessel A. The atmos})heri(' pressures upon the surfaces A and B tAjnd to force the watt-r i '> *Ke two branches of the tube. When the siphon is tilled with water, each of these ])ressu res iu count(Ta(!ted in j)art by the i)rcssure of the water in the branch of the siphon that is immersetl in the water uj)on which the ])ressure is exerted. The atmospheric pressures are very nearly the same for a dilferenci' of level of several feet, owing to the slight /\i.\(i It K Liu ,1 1 t'l ilin»K emptit'd of wj i^cr by tiie ilr forced in 1)V tbe pump. 1 r li' IV arc also contriv ances for tlie expulsion t .f tl le air wlien it iK'Comos impure. 'riie force tending to lift the bell is tlie wei -ht of tbe U' it<>i* (lisulsieod l)V t lie bell mil 1 tbi> cnclnse d >i ir. Hoiiee tbe tension on tbe susptiiding ebain. being equal to tbe weiirbt of tbe bell dimiuLsbcd bv tbe weigbt of water dis- placed by tbe bell ami tbe air witbin, will increase as tbe bell descends, in virtue of tbe diminution of air sjiaco due to tbe increased jirosaure, unless fresb air is forced in from above. Let AlKJD be tbe bell, let EF = (I, tbe deptb of its top below tbe ^ surface of tbe water, FK = /». tbe beijjbt of tbe cvlinder, Fll = i\ tbe ' lengtb occupied by air, t and t' tbe pressures of tbe atmosjiberic air and of tbe comj)ressed air witliiin tbe bell, ami // tbe beigbt of tbe water ba- rometer. Tlien we bave (Art. 48) A F B ( - H 5 V c Fig, 72 7T =: n- X TT -y fjp {a + a;). (1) But, n = gphj wbicb in (1) gives .f2 -f {n -f //) ./; = hb, (« + //) + V\(tT~f^WT^bh (2) tbe positive value only being tbe one wbicb belongs to tbe problem. Coil. — If A ho tbe area of tbe top of tbe bell, and its tbickness be neglected, tbe volume of displaced water is \j\ and tbe tension of tbe cbain ^= weigbt of bell — f/pXx, (:") ! m itw THE COMMOA' PUMP. Sen. — Tho principlo cf \\\v diving hell is appliiMl in div- iiij; (livsst's. 'I'lic (liver is clotht'd in u watcr-liglit dress lilted Willi a helnu'l, and is supplied with air 'y means of a pump. There is an escape valve by wliieli ihi; cireulaliun ^ f fresh air is mainlained. 'I'he diver nuiy he weipfhled up 1(» "^(M) lbs., hul on closin^j; the escape valve, he can rise ul once 1(» Ihe surface in virtue of the buoyancy due tt) the increased displacement, of water hy the enclosed air. 14.'{. TIk^ Common Pump (SiL-tioii Pump).— Any machine* use(l for raising; water from one level to a hif(her, ill which the agency of aiTnos|)heric pressure is employed, is called a pinnp. JMimps are either Kucfioti, furchit/, or liff- imj pumps. r\y N - uffk The pump most <(iinmoiily in use is a sKclinn pumj), of which Fig. 7:J is a ver- tii-al seeliens upwards. Fig* 73 S is a sp«mt a little above A, and C is the surface of the water in which the lower j>art of the pump is immersed. To explain the action of the suet i(m ]Mim[), suppose the piston M to be at H, the pump lilled with ordinary at mos- • Mi'cliiiu'H for raiHiiifj watiT Imvo liceii known from very rarly a^^fH, nnil iln' invention of tlio coniinoii pnnip is ^'cncriilly tiHcrilicil to ('t(*HibiiiK, *<-aclirr of llic celebrated Hero of Alcxundria ; hut the true theory of iiK acttion was not luider- etowl till tbu time uf Qalllcu and TurricvUi. (Sue Suisctuucrti Nai. Phil., p. iXb.) 1" W vv THE ro}f}toy Pt'MP. 209 in (liv- MS of iilat ion |i"i«c al, to tho 'y or, phoric air, and tlic valves V and V (rioscd l»y (heir own weight ; (lie water will staiid al I lie same level (' both within and without the . the pnmp; henci; the at nospherie pressure on the surl'aee of the water outside will force water up the pi|H' HC, until the pressure at C is e AB.BC, or AH BC AB ^ h ' i. e., the ratio of AH to HB must be at least as great as the ratio of BC to h. This condition, although necessary in every case, may not be sufficient. For, suppose that the surface of the water is at Q' when the piston M is at A, in which case the pressure of the air in AQ' = gp {h - Q'C). When the piston descends to H, the pressure in HQ' = gp {h - QC) AQ' HQ' = which must be greater tlian gpli, if the valve is to open, and therefore h'AU > AQ'.Q'C. But the greatest value of AQ'-Q'C is |^AC'; therefore we must have A. AH > iAC'. («) Since fAV^ > AB-BC, unless B is the middle point of AC, it follows that the condition in (2) includes the condi- tion in (1), which is therefore in general insufficient. (See Besaut's Hydrostatics, p. 97.) When ^nd nil. THE LIFTING PUMP. 273 Jn, and ^nce, to below (1) as the sarj in ' when he air , and ifore : of idi- See Fig. 74 146. The Lifting Pump. — When water has to be raised to a height exceeding about 30 feet, the suction pumj) will not work (Art. 143, Sch. 1), and the lifting pump is commonly used. By means of this instrument, water can be lifted to any height. It con- sists of two cyHnders, in the upper of which a piston M is movable, the piston-rod working through an air- tight collar. A pipe DF is carried from the barrel to any required height; at D there is a valve which opens into the pipe. The suction pipe BC is closed by a valve V, as in the suction pump, and the piston M usually * has a valve V. The action of this pump is precisely the same as that of the suction pump in raising Avater from the well into the barrel. Suppose the piston at its higliest point, and the surface of the water in the barrel at K ; then, as the piston is depressed, its valve V will open, und the water will flow throu<^h it till the piston reaches its lowest point. When the piston ascends, lifting the water, the valve D opens, and water ascends in the pipe DF. On the descent of the piston, the valve D closes, and every successive stroke increases the quantity of water in the pipe, until at last it is filled, after which every elevation of the piston will deliver a volume of water equal to that of a cylinder whose base is the area of the piston and whose height is equal to its stroke. The only limit to the height to which water can be lifted is that which depends on the strength of the instrument and the power by which the piston is raised. Cor. — If CK = A, the piston lifts the volume BK at * Sometimes the piston has no valve in it, but is replaced by a solid cylinder, called a plunger, which is operated by a handle us before. 274 TBE FORCING PUMP. M E each stroke, and if yl = the area of the piston, the tension on the piston-rod =. f/pA • BK, until the water is lifted to the valve D, since the air is expelled before the machine is in full action. After this, the power applied to the piston- rod must be increased until the pressure of the water opens the valve D, /. e.y until the pressure =: gp (h + DF), where F is the surface of the water in the tube. The water will then be forced up the tube, the tension of the rod increas- ing as the surface F ascends. 147. The Forcing Pump.— This pump is a further modification of the simple suction pump ; it has no valve in its piston, which is perfectly solid, and works water-tight in the barrel, ranging over the space AE. At the top of the suction pipe BC is a valve, and at the entrance to the pipe DF is a second valve D. When this pump is first set in ac- tion, water is raised from the well as in the common pump, by means of the valve B and piston M, the air at each descent of the piston being driven through the valve D into the pipe DF. When the water has risen through B, the piston, descending, forces it through D ; and when the piston ascends, the valve D closes, and more Avater enters through B. The next de- scent of the piston forces more water through D, and so on until the pipe is filled, as in the lifting pump. The stream which flows from the top of the pipe will be intermittent, as it is only on the descent of the piston that water is forced into the pipe ; but a continuous stream can be obtained bj means of a strong air vessel N (Fig. 76), which consists of a strong brass or copper vessel, at the bot- tom of which is a valve V. Through the top of the air QQS^vQmIQBI B Fifl. 75 Tub FoactNo pvmp. 275 itensiou (ftod to '•ine in pistoii- ^ opens Avhere ter will increas- fnrther falve in valve d de- so on II be that can 76), bot- air Fig. 76 vessel is a discharge pipe KF, which i)asses air-tight nearly io the bottom. WIumi wuler is forced into the Jiir vessel through the valve V by the de- scent of the piston, it rises above the lower end of this pipe. The mass of air which the vessel contains is compressed into a smaller volume; its elas- tic force, pressing on the sur- face of the water at K, with a varying but continuous press- ure, forces it up the pipe ; and if the size of the vessel be suit- able to that of the pump, and to the rate of working it, the com- pressed air will continue to ex- pand, forcing water up the pipe during the ascent of the piston, and will not have lost its force before a new com- pression is applied to it, carrying with it a new supply of water, and thus a continuous, although varying, flow will be maintained. A few strokes of the piston will generally be sufficient to raise water in the pipe _KF, to any height consistent with the strength of the instrument and the power at command. Cor. — Let h = the height of the water barometer; dur- ing the ascent of the piston the valve B is open and V is closed ; the pressure upon the upper surface of the piston = gph ; the pressure upon the lo'/er surface = gp {h — MC), the water surface in the pump being at M ; therefore, call- ing A the area of the piston, the tension of the rod when the piston is ascending = (/pA-M.(J. That is, the tension of the rod is equal bo the iveight of a column of water whose base is the area of the piston, and whose height is the height of the water in the barrel above the level of the well. Sir' 276 SRAMAlfs PRSlS^ m ■■ • m ! 1 ii 148. The Fire Engine.— This is only a modification of the forcing pump with an air vessel, as just described. Two cylinders M and M' are connected with the air vessel V by means of the valves D and D', and the pistons are worked by means of a lever GEG', the ends of which are raised and de- pressed alternately, so that one piston is ascending while the other is descending. Water is thus continually being forced out of the air vessel through the vertical pipe EH, which has a flexible tube of leather attached to it, by means of which the stream can be thrown in any direction. 149. Bramah's Press.*— This press is a practical ap- plication of the principle of the equal transmission of fluid pressures (Art. 8). In the vertical seclion of this instrument (Fig. 78), A and C are two solid pistons or cylinders fitting in air-tight col- lars, and working in the strong hollow cylinders L and K, which are connected by a pipe BD. At D is a valve opening upwards, and at B is a valve opening inwards, a pipe from D communicating with a reservoir of water. M is a mova- ble platform, supporting the substance to be pressed, and N is the top of a strong frame. HOF is the lever working the cylinder C, F being the fulcrum, and H the La'^dle. Action of the Press. — Let C be raised ; the atmospheric * The principle of this press was sugjjested by Stevlnus. It remained unfruitful in practice until 1796, when Bramnh, an English engineer, by an ingenious con- trivance, overcame the only difficulty which prevented its practical application> - Fig. 78 ilAWKSBEE'S AIR-PUMP, %t7 [ficatjon l^„. W pressure forces water from the reservoir through the valve D into the hollow cylinder K, as in the common pump. The cylinder C being pressed down, the valve D closes, and the water is forced through the valve B into L, and, acting on the cylinder A, makes it ascend, thus producing pressure upon any substance included between M and N. A cori- tinued repetition of this process will produce any required compression of the substance. Let R and r be the radii of the cylinders A and C, p the power applied at the handle H, and P the pressure of the water on A j then we have, for the downward force p' on C, , HF But (Art. 9) P'.p' = R^-.r^; (1) m By increasing the ratio of i? to r, any amount of pressure may be produced. Presses of this kind were employed in lifting into its place the Britannia Bridge over the Menai Straits, and for launching the Great Eastern. 160. Hawksbee's Air-Pump.*— B and B' are two cylin- ders, in which pistons P and P', with valves V and V opening upward, are worked by means of a toothed wheel, the one ascend- ing as the other descends. At the lower extremity of the cylinders there are valves v and v' opening upwards, and communicating by means of the pipe AC with the receiver R, from which the air is to be exhausted. Fig. 79 * The air-pump was invented in 1650 by Otto von Ouericke, Burgomaster of Magdeburg. Ifflfl l[l ir! ir !l I 278 HAWKSBEB^S AIR-PUMP. Suppose P at its lowest uiul P' at its highest position, and turn tlie wheel so that P ascends and P' descends. When P' descends, the valve v closes and the air in B' Hows through V, wliile the valve V is closed by the pressure of the external air, and air from R, by its elastic force, opens the v«\lve v and fills the cylinder B. vVhen P descends, the valve V closes, and the air in B being compressed flows through the valve V, while the valve V closes, and. air from the receiver flows through v' into B'. At every stroke of the piston, a portion of the air in the receiver is withdniwn ; and after a considerable number of strokes a degree of mre- faction is attained, which is limited only by the weight of the valves which must be lifted by the pressure of the air beneath. Let A denote the volume of the receiver, and B that of either cyhnder ; p the density of atmospheric air, and Pj, Pg, pn the densities in the receiver after 1, 2, n descents of the pistons. Then after the first stroke tlie air which occupied the space A will occupy the space A + B, and therefore we have Pi {A + B) z=z pA. Similarly, pg {A -\- B) = p^A ; .-. pg {A + Bf = pA% and after n strokes we have Pn{A + B)n = pA% the volume of tlie connecting pipe AC being neglected. Hence, calling tt^ and n the pressures of the air in the receiver after n strokes and of the atmospheric air respect- ively, we have ^ _Pn _ I A Y n\ T^ ~ p ~\A -{• b)' ^ r Thus, suppose that A is four times B, and we were re- 8MEAT0N S AIR-PUMP. 279 quired to find the density of the uir in tlie receiver at the end of the loth stroke, we iiave from (1) If the air originally had an elastic force equal to the pressure of 30 in. of mercury, this would give the elasHc force of the air remaining in the receiver as equal to a pressure of 1.05G in. of mercury. In this case, it is custom- ary to say that the vacuum pressure is one of 1,050 in, of mercury. ScH. — It is evident from (1) that p„ can never become zero as long as n is finite, and therefore, even if the machine were mechanically perfect, we could not by any number of strokes completely remove the air; for, after every stroke there would be a certain fraction left of that which occupied it before. In working the instrument, the force required is that which will overcome the friction, together with the differ- ence of the pressures on the under surfaces of the jiistons, the pressures on their upper surfaces being the same, 151. Smeaton's Air-Pump.— This instrument con- sists of a cylinder AB in which a piston is worked by a rod passing through an air-tight collar at the top ; a pipe BD passes from B to the glass receiver C, and three valves, open- ing upwards, are placed at B, A, and in the piston. Suppose the receiver and cylinder to be filled with atmospheric air, and the piston at B. Raising the piston, the valve A is opened by the compressed air in AM which flows out through it, while at the same time a portion of the air in C flows through the pipe DB to fill the partial vacuum formed in MB, so that when the juston arrives at A, the air which at first occupied C now fills both M ff Ti 'i i i m r\ IID Fig. 80 280 THE HYDRAULW RAM. ■ '■W the receiver and the cylinder. Wlien the piston descends, the valves A and B close, and the air in the cylinder below the piston is compressed until it opens the valve M, and passes above the piston. As the piston is raised a second time the valve A is opened by the compressed air in AM, which flows out through it as before; and thus at each stroke of the piston a portion of the air in the receiver is forced out through A. Let A and B denote the volumes of the receiver and cylinder respectively, and p and pn the densities of atmos- pheric air and of air in the receiver after w strokes. Then, as in Art. 150, we have Pn{A -^ BY = pA\ from which it appears as in the previous article that, although the density of the air will become less and less at every stroke, yet it can never be reduced to nothing, however great n may be. ScH. — An advantage of this instriiment is that, the iipper end of the cylinder being closed, when the piston descends the valve A is closed by the external pressure, and therefore the valve M is then opened easily by the air beneath. Also the labor of working the piston is diminished by the removal, during the greater part of the stroke, of the atmospheric pressure on M, which is exerted only during the latter part of the ascent of the piston, when the valve A is open. \ 152. The Hydraulic Bam."^ — The hydraulic ram is a machine by which a fall of water from a small height produces a momentum which is made to force a portion of the water to a much greater height. Fig. 81 * Invented by Montgolfier. ■ THE HYDRAULIC RAM. 281 sends, Ibelow r, and lecoiid AM, each Ivor is In the vertical section (Fig. 81), AB is the descending and FG the ascending column of water, which is sup- plied tiom a reservoir at A. V is a valve opening down- wards, and V is a valve opening upwards into the air- vessel C ; H is a small auxiliary air-vessel with a valve K opening inwards. T%e Action of the Machine. — As the valve V at first lies open by its own weight, a portion of the water, descend- ing from A, flows through it ; but the upward flow of the water towards the valve V increases the pressure tending to lift the valve, and at last, if the valve is not too heavy, lifts and closes it. The forward momentum of the column of water ABD being destroyed by the stoppage of the flow, the water exerts a pressure sufficient to open the valve V and to flow through it into the air-vessel C, condensing the air within ; the reaction of the condensed air forces water up the i)ipo FG. As the column of water ABD comes to rest, the pressure of the water diminishes, and the valves V and V both fall. The fall of the former produces a rush of the water through the opening V, followtA by an increased flow down the supply pipe AB, the result of which is again the closing of V, and a repetition of the process just described, the water ascending higher in FG, and finally flowing through G. The action of the machine is assisted by the air-vessel H in two ways — first, by the reaction of the air in H, which is compressed by the descending water, and, secondly, by the valve K, which affords supplies of fresh air. When the water rises through V, the air in H suddenly expands, and its pressure becoming less than that of the outer air, the valve K opens, and- a supply flows in, which compensates for the loss of the air absorbed by the water and taken up the column FG, or wasted through V. About a third of the water employed is wasted, but the machine once set in motion will continue in action for a long time, provided the 282 WORK OF WATER WHEELS. t I ! J supply in the reservoir be maintained. (See Besant's Hydrostatics, p. 112.) 153. Work of Water Wheels.— To utilize ahead of water, consisting of an actual elevation above a datum level at whicli the water can be delivered and disposed of, a machine may be employed in which the direct action of the weiglit of the water, while falling through the given height is the principal moving force. When a stream of water strikes the paddles of a wheel which has a certain velocity, the energy imparted to the wheel by the water, from (4) of Art. 98, = [^-iy- vn ^, (1) where V is the velocity of the periphery of the wheel, v the original velocity of the water, and W the weight of water acting on the wheel ])er second ; but if the water descends with the puddle there is an additional amount of work done on the wheel dne to the mean height h throngh which the water falls. Hence we have, for the whole work done on the wheel per cccond, = [,.2 _ (y _ r)2J ^ + PF/^ (2) Now if the water leaves the paddles the work remaining in the water will be lost ; hence, calling r, the velocity of the water after it has left the paddles, we have for the use- ful work U done on the wheel W U=\f-{v-VY- Vi^] ^ + Wh = [2vV- V'i — Vi»] ^ + Wh, (3) i i which is the general expression for the work done by a water wheel when the water impinges upon the paddles; IKjrpendicularly. r WORK OF OVERSHOT WHEELS. 283 l^ad of level of, a [on of given (1) (2) (3) 164. Work of Overshot Wheels.— When a waterfall ranges between 10 and 70 feet, and the water supply is from 3 to 25 cubic feet per second, it is possible to construct a bucket wheel on which the water acts chiefly by its weight. If the varia- tion of the head-water level does not exceed 2 feet, an overshot wheel may be used. The water is then projected over the summit of the wheel, and falls in a parabolic path into the bucket. If v be the velocity of delivery to the wheel, the part— is converted into energy of motion before reach- ing the buckets and operates by impulse ; hence in a wheel of this class the water does not operate entirely by weight. The height h through which the water falls is the vertical height of the point at which the water meets the buckets above the point where it leaves them, Avhich in this wheel is nearly equal to the diameter of the wheel ; and as the velocity of the water on leaving the bucket is the same as the velocity of the bucket itself, we have v^ =V\ hence (3) of Art. 163 becomes mmm^^^^mPTPm U={v-V)V^ + Wh, (1) Galling 9?t tho efficiency* of these wheels, we have from (1) U=m\ i(v-F)F-f a1 W. m Cor. — To find the relation of v and V so that the useful work V of the wheel may be a nuiximum, we must equate to zero the derivative of U with respect to V, which gives <■ S^e Mai. Mechs., AH. 916. 284 WORK OF BREAST WHEELS. V = ^v, i. e., the wheel works to the best advantage when the velocity of its periphery is one-half that of the stream. ScH. — If the velocity of the periphery of this wheel is too great, water is thrown out of the buckets before reach- ing the bottom of the fall. In practice, the circumferential velocity of water wheels of this kind is from 4|^ to 10 feet per second, about 6 feet being the usual velocity of good iron wheels not of very small size. The velocity of the water therefore is limited to about 12 feet per second, and the part of the fall operating by impulse is therefore about 2^ feet. The rest of the fail operates by gravitation, but a certain fraction is wasted by spilling from the buckets, and emptying them before reaching the bottom of the fall. The great diameter of wheel required for very high falls is in- convenient, but there are examples of wheels 60 feet in diameter and more. The efficiency of these wheels under favorable circum- stances is 0.75, and is generally about 0.65. 155. Work of Breast Wheels. — When the variation of the head-water level exceeds 2 feet, a breast wheel is better than an overshot. In breast wheels the buckets are replaced by vanes which move in a channel of masonry partially surrounding the wheel. The water falls over the top of a slid- ing sluice in the upper part of the channel. The channel is thus filled with water, the weight of which rests on the vanes and furnishes the motive force on the wheel. There is a certain amount of leakage between the vanes and the sides of the channel, but this loss is not so great as that by spilling from the buckets of the overshot wheel, WORK OF UNDERSHOT WHEELS. IvdS itage It of eel is •each- ntial Ofeet good f the , and about but a , and The IS in- eet in In this wheel, as in the case of the overshot wheel, v, = F, therefore (1) and {Z) of Art. 154 also apply to breast wheels, h being the height of the point at which the water meets the vanes above the point where it leaves them. The efficiency is found by experience to be as much as 0.75. ScH. 1. — Theoretically this wheel also works to the best advantage when the speed of its periphery is one-half that of the stream (Art. 154, Cor.). But Morin found, by ex- periments, that the efficiency of the wheel is not much affected by changes in its velocity. This is owing to the circumstance that the useful work is dependent principally upon the term Whj and not upon the other term in the formula which alone is affected by the velocity of the wheel. Hence the great advantage of this wheel is, that it may be worked, without materially impairing its efficiency, with velocities varying from |y to f v. ScH. 2. — As the diameter of this wheel is greater than the fall, a breast wheel can be employed only for moderate falls. Overshot and breast wheels work badly in back-water, and hence if the tail-watei* level varies, it is better to reduce the diameter of the wheel so that its greatest immersion in flood is not more than one foot. 156. Work of Undershot Wheels. — The common un- dershot wheel consists of a wheel provided with vanes, against which the water im- pinges directly. In this case tlie water is allowed to attain a velocity due to a considera- ble part of the head immediately before entering the ma- chine, so that its energy is nearly all converted into energy of motion ; and as the water has no fall on the wheel, and i : 286 WOSK OF THE FQNCELET WATER WHEEL. its velocity on leaving the vanes is the same as the velocity of the vane itself, we have // = 0, v^ =V; therefore (3) of Art. 153 becomes W or U = m {v— V)V W g' (1) (8) where m, as before, is the efficiency of the machine. ScH. — The wheel works to the best advantage when the speed of the periphery is one-half that of the stream (Art. 154, Cor.), but the efficiency is low, never exceeding 0.5. Wheels of this kind are cumbrous. In the early days of hydraulic machines, they were often used for the sake of simplicity. In mountain ceun tries, where unlimited power is available, they are still found. The water is then con- ducted by ab artificial channel to the wheel, which some- times revolves in a horizontal plane. When of small diameter, their efficiency is still further diminished.* 167. Work of the Poncelet Water Wheel.— When the fall does not exceed G feet, the best water motor to adopt in many cases is the Poncelet undershot water wheel. In the common undershot water wheel, the paddles are flat, whereas in the Poncelet wheel they are curved, so that the direction of the curve at the lower edge, where the water first meets the paddle, is the same as the direction of the stream. By this arrangement, the water, which is allowed to flow to the wheel with a velocity nearly equal to the velocity due to the whole fall, glides up the curved floats without meeting with any sudden obstruction, comes to relative rest, then descends along the float, and ac{iuires at the point of discharge from the float a backward velocity relative to the wheel nearly equal to the forward velocity of * See Cotteriiri App. Meche. ; also, Fairbairn'a Millwork and Machinery. WORK OF THE PONCELET WATER WHEEL. 287 )city (3) (1) (2) the wheel. The water will therefore drop off the floats de- prived of nearly all its kinetic energy. Nearly the whole of the work of the stream must therefore have been expended in driving the float ; and the water will have been received without shock, and discharged without velocity. Let V and V be the velocities of the stream and float re- spectively ; then the initial velocity of the stream relative to the float is v — V, and the water will continue to run up the curved float until it comes to relative rest ; it will then descend along the float, acquiring in its descent, under the influence of gravity, the same relative velocity which it had at the beginning of its ascent, but in a contrary direction. Therefore the absolute velocity of the water leaving the float is V—{v — V) = 2V-v. Now the useful work U done on the wheel must equal the work stored in the water at first, diminished by the ^ork stored in the water on leaving the wheel j hence W W i^lV—vf 9 Comparing this expression with (1) of Art. 156, we see that the work performed by the Poncelet wheel is double that of the common undershot wheel. ScH. — This wheel works to the best advantage when the speed of the periphery is one-half that of the stream (Art. 154, Cor.). This conclusion also follows from the form of the floats, as above described ; since if all the work is taken out of the water when it leaves the floats, its velocity must then be zero, and therefore 3 F— ?» = 0, or F=r |i;.* The efficiency of a Poncelet wheel has been found in ex- * The inventor, Poncelet, states that, In practice, the velocity of the water, in order to produce its maximum effect, ought to he about 21 times that of the wheel, And that the efficiency of the wheel Is about 0.7 (Tate'e Mecb, Fhil., p. 818). m !l^ 288 TSE HE ACTION WHEEL J BARKER'S MILL. Fig. 85 periraents to reach 0.68. It is better to take it at 0.6 in estimating the power of the wheel, so as to allow some margin. 158. The Reaction Wheel; Barker's Mill.— Fig. 85 shows a simple reaction wheel. AOB is a tube* capable of revolving about its axis, which is vertical, and having a horizontal tube DBE connected with it. Water is sup- plied at C, which descends tlirough the vertical tube, and issues through the ori- fices D and E at the extremities of the horizontal tube, so placed that the direc- tion of motion of the water is tangential to the circle described by the orifices. The efflux is in opposite directions from the two orifices ; as the water flows through BD, the press- ures on the sides balance each other except at D, where there is an uncompensated pressure on the side opposite the orifice ; the effect of this pressure or reaction is to cause motion in a direction opposite to that of the jet. The same effect is produced by the water issuing at E, and a continued rotation of the machine is thus produced by the reaction of the jet in each arm. Let h be the available fall, measured from the level of the water in the vertical pipe to the centres of the orifices, v the velocity of discharge through the jets, and V the veloc- ity of the orifices in their circular path. When the machine is at rest, the water issues from tlie orifices with the velocity V^gh (neglecting friction). But when the machine ro- tates, we have for the velocity of discharge through the orifices, from (1) of Art. 89, V V T^ + 2gh. (1) While the water passes through the orifices with the ve- locity V, the orifices themselves are moving in the opposite THE REACTION WHEEL; BARKER'S MILL, 289 ).6 in I some 11. - I tube; direction with the velocity V. The absolute velocity of the water is therefore v—V=z^/V^+'^gh-V. (2) Now the useful work done per second by each pound of water must equal the work due to the height h, diminished by the work remaining in the water after leaving the machine. Hence, (v — F)2 useful work = h — - — ^ — — 2g _{VV^^2gh-V)V ~ g , from (2), (3) The whole work expended by the water fall is h foot- pounds per second ; consequently, to find the efficiency of the machine, we divide (3) by h (Anal. Mechs., Art. 216), and get efficiency = (VJ^+MzlZII (5) gh gh + etc. 2V' (by the Binomial Theorem), (6) which increases towards the limit 1 as V increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize ; and even at practicable but high velocities of rota- tion, the prejudicial resistances, arising from the friction of the water and the friction upon the axis, would considera- bly reduce the efficiency. Experiment seems to show that the best efficiency of these machines is reached when the velocity is that due to the head, so that V^ = 2^A. 4 290 THE VENTlilFUOAL PUMP, Wl When V^ = llgh, we have, from (5), neglecting friction, (7) efficiency = V V 2 1) F^ _ ^ g^g^ about 17 per cent, o'f the energy of the fall being carried away by the water discharged. The actual efficiency real- ized of these machines appears to be about 60 per cent., so that about 32 per cent, of the whole head is spent in over- coming frictional resistances, in addition to the energy carried away by the water. ScH. — The reaction wheel in its crudest form is a very old machine known as " Barker^s Mill." It has been em- ployed to some extent in practice as an hydraulic motor, the water being admitted below and the arms curved. In this case the water is transmitted by a pipe which descends be- neath the wheel and then turns vertically upwards. The vertical axle is hollow, and fits on to the extremity of the supply pipe with a stuffing box. In this construction the upward pressure of the water may be made equal to the weight of the wheel, so that the pressure upon the axis may be nothing. These modifications do not in any way affect the principle of the machine, but the frictional resistances may probably be diminished. 169. The Centrifugal Pump.— "When large quanti- ties of water are to be raised on a low lift, no pump is so suitable as a centrifugal pump. In this pump, water is raised by means of the centrifugal force given to the water in a curved vane or arm, proceeding from the vertical axis. The dynamic principles of this machine are the same as those of the reaction wheel (Art. 158) ; but they differ in tlieir objects. In the latter machine, a fall of water gives a rotatory motion io a vertical axis, while in the former a rotatory motion is given to a vertical axis in order to ele- vate a column of water. uri of th( pa fo] he efl ai 3 THE CENTRIFUGAL PUMP. 291 (7) ried [eal- L so |ver- Let h be the height to which the water is raised, meas- ured from the level of the water in the well to the centre of the orifice of discharge, v the ?locity of discharge through the orifice, and V the velocity of the orifice in its circular path, as in Art. 158. Then the work due to the centrifugal force must equal the work of raising the water through the height //, increased by the work stored in the water at efflux ; therefore -■5 — = w^ + "s— ; .-. V = V F8 — 2^A, (1) and • [as in (2) of Art. 158]. Now the work applied per second to raise each lb. of water must equal the work in raising the water through the height //, increased by the work remaining in the water after leaving the machine. Hence applied work = A + {v - Vf (Y-.^V^ — 2gh)V 9 (3) The useful work is h foot-pounds per second ; therefore gh efficiency = = 1 {V-\/V^-'-Zgh)V gh 2F2 — etc.. (3) (4) which increases towards the limit 1 as F increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the pump has an infinitely great velocity of rotation, as in the case of the reaction wheel. 292 TURBINES. CoR.~When V^ = 2gh, we have, from (3), efficiency = 0.5. When F' = ^gh, we have, from (3), . 1 efficiency = ■:=: = .86. 2 (2 - \/2) When F* = Qgh, we have, from (3), efficiency = 0.9. Hence, theoretically, the centrifugal pump has a con- siderable efficiency ivhen the velocity of rotation exceeds the velocity due to twice the height of the col- umn of waiter raised. ScH. — Centrifugal numps work to the l9st advantage only at the particular lift for which they are designed. When employed for variable lifts, as is constantly the case in practice, their efficiency is much reduced, and does not exceed .5, and is often much less. The earliest idea of a centrifugal pump Avas to employ an inverted Barker's Mill, consisting of a central pipe dipping into water, connected with rotating arms placed at the level at which water is to be delivered. The first pump of this kind which attracted notice was one exhibited by Mr. Ap- pold in 1851, and the special features of this pump have been retained in the best pumps since constructed. The experiments conducted at the Great Exhibition on Appold's Centrifugal Pump with curved arms, gave the maximum efficiency 0.68. But when the arms were straight and ra- dial, tlie efficiency was as low as .24, showing the great advantage of having the curved form of the arms, which causes the water to be projected in a tangential direction. 160. Turbines. — A reaction wheel is defective in prin- ciple, because the water after delivery has a rotatory veloc- ity, in consequence of which a large part of the head is TURBINES. 203 wasted (Art 158). To avoid this, it is necessary to employ a machine in which some rotatory velocity is given to the water before entrance, in order that it may be possible to discharge it with no velocity except that which is absolutely required to pass it through the machine. Such a machine is called a Ttrbine, and it is described as ** outward flow," "inward flow," or "parallel flow," according as the water during its passage through the machine diverges from, con- verges to, or moves parallel to the axis of rotation.* Turbines are wheels, generally of small size compared with water wheels, driven chiefly by the impulse of the water. The water is allowed, before entering the moving part of the turbine, to acquire a considerable velocity ; dur- ing its action on the turbine this velocity is diminished, and the impulse due to the change of momentum drives the turbine. Koughly speaking, the fluid acts in a water-pressure engine directly by its pressure ; in a water wheel chiefly by its weight causing a pressure, but in i)art by its kinetic energy, and in a turbine chiefly by its kinetic energy, which again causes a pressure, f In the outward and inward flow turbines, the water en- ters and leaves the turbine in directions normal to the axis of rotation, and the paths of the molecules lie exactly or nearly in planes normal to the axis of rotation. In outward- flow turbines the general direction of flow is away from the axis, and in inward-flow turbines towards the axis. In parallel-flow turbines, the water enters and leaves the tur- bine in a direction parallel to the axis of rotation, and the paths of the molecules lie on cylindrical surfaces concentric with that axis. There are many forms of outward-flow turbines, of which the best known was invented by Fourneyron, and is com- monly known by his name. The inward-flow w:is invented by Prof. Jas. Thomson. * Cotterill^'s App. Hechs., p. 606, t Bncy. Brit,, Vol. ^H., p. 58Q. 294 EXAMPLES. The theory of turl)ines is too intricate a subject to be considered in this treatise. For a general classification of turbines, with descriptions, illustrations, and discussions of these machines, as well us for a further development of hydraulic machines in detail, the student is referred, among other treatises, to the following : Fairbairn's Millwork and Machinery, Colyer's Water-Pressure Machinery, Barrow's Hydraulic Manual, Glynn's Power of Water, Prof. Unwin's Hydraulics. EXAMPLES. 1. In a hydrostatic bellows (Fig. 70), the .ube A is | of an inch in diameter, and the area DE is a circle, the diam- eter of which is a yard. Find the weight wliich can be supported by a pressure of 1 lb. on the water in A. Ans. 82,944 lbs. 3. Describe tiie siphon and its action. What would be the effect of making a small aperture at the highest point of a siphon ? 3. A prismatic bell is lowered until the surface of the water within is 6G feet below the outer surface; state approximately how much the air is compressed. Ans. To ^ of its original volume. 4r If a prismatic bell 10 feet high be sunk in sea water until the water rises half way up the bell, find how far the top of the bell must sink below the surface, the tempera- ture remaining the same. Assume the water barometer — 33 feet for sea water. Ans. 28 feet. 5. In the jmsition of the bell in Ex. 4, find how much air must be forced into it in order to keep the water down to a level of 2 feet from its bottom. Ans. 0.72 W, where W is the weight of the air in thQ bell when at the surfa«?i EXAMPLES. 295 to be |n of 18 of It of long und fow's 'in's 6. If a small hole be made in the top of a diving bell, will the water flow in or the air flow out ? 7. If a cylindrical diving bell, height 5 feet, be let down till the depth of its top is 55 feet, find (1) the space occupied by the air, and (2) the volume of air that must be forced in to expel the water completely, the water barometer standing at 33 feet. Ans. (1) 1.8 nearly ; (2) f Jths of the volume of the bell. 8. The weight of a diving bell is 1120 lbs., and the weight of the water it would contain is G72 lbs. Find the tensioi. of the rope when the level of the water inside the bell is 17 feet below the surface {h = 33 feet). Ans. 670.48 lbs. 9. A cylindrical diving bell of height a is sunk in water till it becomes half full. Show that the dej)th from the surface of the water to the top of the bell is h — - • 10. A cylindrical diving bell, of which the height inside is 8 ft., is sunk till its top is 70 feet below the surface of the water. Find the depth of the air space inside tlie bell {h = 33 feet). Ans. 2| feet. 11. (1) Describe the action of a common pump ; (2) distinguish between a lifting pump and a forcing pump ; (3) to what height could mercury be raised by a pump? 12. The length of the lower pipe of a common pump above the surface of the water is 10 feet, and the area of the up])er pipe is 4 times that of the lower ; prove that if at the end of the first stroke the water just rises into the upper pipe, the length of the stroke must be 3 feet 7 inches very nearly {h = 33 feet). 13. If the diameter of the piston be 3 inches, and if the height of the water in the pump be 20 feet above the well, what is the pressure on the piston ? Ans. 61.^ Ibg. 296 EXAMPLES. 14. If the diameter be 3 J- inches, the height of the water in the pump 27 feet 5 inches, the lever handle 4 feet, and the distance from the fulcrum to the end of the piston rod 4 inches, find the force necessary to work the pamp*handle. Ans. 9| lbs. 15. The height of the column of water is 60 feet above the well, the piston has a diameter of 3 inches, the pump- handle is 3^ feet from the fulcrum, and the distance of the fulcrum from the piston rod is 3^ inches ; find the force necessary to work the pump. Ans. 15.3 lbs. 16. If the height of the cistern above the well be 25 feet, the diameter of the piston 3 inches, and the leverage of the handle 12 : 1, find the force necessary to use in pumping. Ans. 2.83 lbs. 17. If the height of the cistern be 42 feet, the diameter of the piston A^ inches, the length of the handle 49 inches, and the distance of the fulcrum from the piston rod 3^ inclies, find the force. Ans. 20.65 lbs. 18. The diameter of the piston of a lifting pump is 1 foot, the piston range is 'H^ feet, and it makes 8 strokes per minute; find the weight o." water discharged per minute, supposing that the highest level of the piston range is less than 33 feet above the surface in the reservoir (A = 33 feet). Ans. 312.57r lbs., or about 983 lbs. 19. If in working the pump of Ex. 18, the lower level of the piston range be 31^ feet above the surface in the reser- voir, find the weight of water discharged per minute. Ans. 187. 5Tr lbs. 20. In a Bramah's press FO = 1 inch, FH = 4 inches, the diameter of yl = 4 inches, and diameter of C = ^ an inch ; find the force on A produced by a force of 2 lbs. applied at H* , Ans, 512 lbs. EXAMPLES, 297 21. In one of the Bramah presses used in raising the Britannia tube over the Menai Straits, the diameter of the piston C was 1 inch, that of A 20 inches; th, force applied to C at each stroke was 2| tons ; find the lifting force pro- duced by the upward motion of A. Ans. 1000 tons. 22. If the receiver be 4 times as large as the barrel of an air-pump, find after how many strokes the density of the air is diminished one-half. Ans. Early in the 4th stroke. 23. After a very great number of strokes of the piston of an air-pump the mercury stands at 30 inches in the barometer-gauge, the capacity of the barrel being one- third that of the receiver, prove that after 3 strokes the height of the mercury is very nearly 12f inches. 24. A fine tube of glass, closed at the upper end, is inverted, and its open end is immersed in a basin of mer- cury, within the receiver of a condenser ; the length of the tube is 15 inches, and it is observed that after 3 descents of the piston the mercury has risen 6 inches; how far will it have risen after 4 descents ? 15 Ans. The ascent x is given by the equation 20 lb — x^ h = \ + Sh If A = 30, X =6.1 nearly. 25. It A = SB (Art 151), find the elastic force of the air in the receiver after the 5th, 10th, 15th, and 20th strokes, the height of the barometer being 30 inches. Ans. 7.119 ins.; 1.689 ins.; 0.401 ins.; 0.095 ins. 26. In the same pump, the barometer standing at 30, find the number of strokes, (1) when the mercury in the gauge rises to 25 inches, and (2) when the rarefaction is l-^100. A71S. (1) 6.2; (2) 16. 27. If a hemispherical diving bell be sunk in water until the surface of the water inside the bell bisects its vertical 298 EXAMPLES. \i 1 radius, find tlie depth of the bell, supposing the atmos- pheric pressure to be .14.28 lbs. to the square inch {Ji = 34). Jna. From surface to surface 73.3 feet. 28. There is a pump lifting water 29 feet high, tlie diameter of its piston is 1 foot, the play of the piston is 3 feet, and the pump makes 10 strokes per minute ; (1) how many gallons of water will be discharged per minute, and (2) what is the pressure on the piston ? Ans. (1)147 gals.; (2) 1420 lbs. 29. Water flowing through a trough, 2 'eet wide and 1 foot deep, with a velocity of 10 feet per second falls upon an overshot wheel 50 feet in diameter. Find (1) the part of the fall operating by impulse; (2) the maximum useful work of the wheel, the efficiency being 0.70 ; and (3) the number of revolutions the wheel makes per hour when doing maximum work. Ans. (1) 1.55 feet ; (2) 43076.25 ft.-lbs. per sec. ; (3) 114.o9. 30. Water is furnished to a breast-wheel at the rate of 20 cubic feet per second with a velocity of 8 feet. The fall is 20 feet and the efficiency 0.75. What is the useful work done by the wheel when the periphery has a velocity of 3, 4, and 5 feet per second respectively ? (See Sch. 1, Art. 155). Ans. 19185.94; 19215.94; and 19185.9 ft.-lbs. per sec. respectively. 31. What is the useful work done by an undershot wheel, 40 feet in diameter, making 120 revolutions per hour, the velocity of the water being 20 feet per second and the area of the vanes being 1| square feet ? Ans. 1910 ft.-lbs. per second. 32. What is the efficiency of a reaction wheel when the water having a head of IG feet, issues from the orifices with a velocity of 45 feet per second ? Ans. 0.8254. the atnios- 3li {h = 34). 73.3 feet. t high, the 9 piston is 3 te ; (1) how ninute, and U20 lbs. wide and 1 falls upon 1) the part lum useful nd (3) the hour when sec; (3) the rate of The fall seful work ocity of 3, Jh. 1, Art. . per sec. undershot tions per icond and second. when the fices with 0.8354.