UC-NRLF SB 77 v V '<. ~^ UNIVERSITY OF CALIFORNIA ANDREW SMITH HALLIDIL: VMtoK r z, w being the intrinsic weight of the fluid and 2 the difference of level. The free surface of a liquid at rest under gravity is a horizontal plane.. The common surface of two liquids of different densities, which do not mix, is a horizontal plane, when at rest under gravity. If a number of liquids of different densities, e.g., mercury, water.and oil, are poured into a vessel, they will come to rest with their common surfaces horizon- tal planes, the densities of the liquid increasing downwards. The surfaces of equal pressure are horizontal planes. The pressure of a liquid on any horizontal area, A, is equal to the weight of a column of the liquid of which the area is the base and of which the height, z, is equal to the depth of the area below the surface, i.e., wAz (disregarding the pressure on the free surface). The whole pressure of a fluid on a submerged surface is the sum of all the normal pressures exerted by the fluid on every portion of the sur- face and (disregarding the pressure on the free surface) is equal to the weight of a column of liquid of which the base is equal to the area of the surface, and the height is equal to the depth of the centroid of the surface below the surface of the liquid. Thus : (a) The total normal pressure on a wall of width b, sloping at 6 to the vertical and retaining water which rises over a length z of the wall z ivb2* cos = ivbz cos --- ' (b) The total pressure on a circular valve of diameter d, with its cen- TtlP troid z below the surface = w z. 4 (0 The total normal pressure on a lock-gate of width b and on which the water rises to a height z = wbz^ ivbz' 1 . The pressure between a pair of lock-gates = pressure on the hinge post = wbz* sec a, ia being the angle between the gates. 'The centre of pressure of a plane area is the point of action of the resultant fluid-pressure, (/?) upon the plane area. If y, ~z are the horizontal and vertical distances of the C. of P. from the vertical and horizontal axes through the C. of G. of the area, _ wD iv D _ D - _ ool _ 9 ~~ ~X~ = ^Ah = ~Ah ' a ~~ R ~ D being the product of inertia about the axes ; /the moment of inertia of the area about the axis of y ; // the depth below the surface of the centroid, and k the radius of gyration. HYDROSTATIC PRINCIPLES. xv Ex. i. Depth of C. of P. of a parallelogram with one edge in surface .= f of depth of opposite edge. Ex. 2. Depth of C. of P. of a triangular area, the middle points of the sides being at depths d\, di, d* below the surface, = - - -- , tfi + rta + a* and (a) if vertex is in surface and base horizontal, depth = f of depth of base ; (#) if base is in surface, depth = | of depth of vertex ; (r) i/ vertex is in surface and y and z are depths of ends of base, the i y 2 3 depth W^-y. The resultant pressure on the surface of a solid, wholly or partially immersed in a fluid, is equal to the weight of the displaced liquid and acts vertically upwards in a line passing through the centroid of the dis- placed liquid. In other words, a solid immersed in a liquid appears to lose as much of its weight as is equal to the weight of the fluid it displaces. If a homogeneous body float in a liquid, its volume will bear to the volume immersed the inverse ratio of the specific gravities of the solid and liquid. A body of weight W, carrying a load P, floats in a liquid, G and h being the centres of gravity of the body and of the displaced water, so that GH is vertical. If the load P is shifted, the body will heel through an angle and the point H, also called the centre of buoyancy, will move on a curve or surface of buoyancy to a new position H' , the line G'H connecting H' with the new position of the C. of G. of the body being vertical. If 6 is small, the ultimate position of M, the intersec- tion of HG and H' G' , is called the metacentre, and M is therefore the centre of curvature of the surface of buoyancy at H. For stability of equilibrium M must be above G. Theoretically, iv I wAk* A& HM -TJJ- = - =7- = - - , W wV V A being the water-line area and F the volume of liquid displaced by body. CAPILLARY PHENOMENA. If a glass tube of fine bore is placed verti- cally in a liquid like water, which wets the glass, the water-surface on the outside next the glass is elevated and slightly concave, while on the inside the water-surface is concave and there is a marked elevation above the outside surface. With a liquid which does not wet the glass, like mercury, an opposite effect is observed. There is a depression on the outside and the surface xvi HYDROSTATIC PRINCIPLES. is slightly convex, while on the inside the surface is convex and there is a marked depression below the outside surface. SURFACE TENSION. At the bounding surface separating air from any liquid, or between two liquids, there is a surface-tension which is, the same at every point and in every direction. At the line of junction of the bounding surface of a gas and a liquid with a solid body, or of the bounding surface of two liquids with a solid body, the surface is inclined to the surface of the solid body at a definite angle, depending upon the nature of the solid and the liquids. The surface-tension is independent of the curvature of the surface but, if the temperature be increased, it diminishes. USEFUL CONSTANTS. The following abbreviations are used: Metre = m.; sq. metre = m.*; cubic metre = m. 3 ; centimetre = cm.; sq. centimetre = cm. 1 ; cubic centi- metre cm. 3 ; kilometre kilo.; grain = gr. ; gramme = gm.; kilo gramme k.; kilogramme metre = km. i British ton 2240 Ibs. = 1016 k. i U. S. ton = 2000 Ibs. = 907-143 k- I in. 2.54 cm. i cm. = -39370II in. I ft. = 30.4709 cm. i m. 3.280843 ft. i mile = 1.6093 kilo. I kilo. = .62137 mile. i knot = i naut. mile per hr. =r 6080 ft. (av.) per hr. i sq. in. = 6.4516 cm. 2 i cm. 9 = .155 sq. in. i sq. ft. = 929.03 cm. 1 i m. 2 = 10.7639 sq. ft. i sq. yd. . 836126 m.* I acre = 43.56osq. ft. = .40468 hectare. i hectare = 10,000 m. f = loo ares. 2.4711 acres. I sq. mile = 640 acres. 2.59 sq. kilo. = 259 hectares. I sq. kilo. == 100,000 m. 2 = 24.711 acres. I Ib. 16 oz. = 7000 gr. = -4535924 k. = 453- 5924 gm. = 445,000 dynes. Ik. = 2 204622 Ibs. = 981,000 dynes. i Fr. tonne = loco k. = .9842 British ton. = 2204.622 Ibs. I cu. in. of water at 4 C. = 252.89 gr. i cm.' " = i gm. i cu. ft. " " = 62.43 Ibs. i litre " " i k. i imp. gal. at 62 F. = 10 Ibs. I cu. ft. of water at 62 F. = 62.3 Ibs. i cu. ft. of air at o C. and i atm. = .0807 Ib. I cu. ft. of hydrogen at ) o C. and . atm. {"""SHS. i litre of air at o C. and i atm. Water compresses of its bulk under a change of pressure of i atm., or about -^th of its vol. un- der a pressure of 2 tons (of 2240 Ibs.) per sq. in. i Ib, per sq. in. = .0703 k. per cm. 2 I k. per cm. 2 = .0703 lb.persq.in. i Ib. per sq. ft. = 4.8826 k. per m. 2 = 479-dynespercm. 2 xvii xviii USEFUL CONSTANTS. No. of Ibs. per ) sq. in. j" = 14.223 k. per cm. i standard atm. 1 ins of ofi 4 . 7 lbs.per^| me ( .4907 ins. of ~ I mercury. sq. in. J = 760 mm. i metric atm. of 1 j ins. of mercury ~ \ -T- 2.0378. -, ( 28.96 ins. of 14. 223 Ibs. per }- = 3 sn in ' < mercury. No. of k per m. s _ j 4.8826 Ibs. per sq. in. - J i erg = i dyne X i cm. ~ ( sq. ft. i gm.-cm. = 981 ergs. i in. of mer- ) cury at o C. ) = . 034534k. per cm. 2 i ft.-lb. = .13825 km. = 1-3562 Xio 7 ergs. I mm. mercury j . o y = .0013596 per cm. 2 i km. = 7.233 ft.-ibs. at o C. ) = 9.81 X io 7 ergs. No. of ft.-lb. =7.2178 km.. i cu. in. = 16.387 cm. s = 777 B. T. U. i cm. 3 I CU. ft. = .061 cu. in. = .028317 m. J _ ( 1399 Ibs. de- ( gree C. = 28.317 litres. i B. T. U. = 1058 joules. i m 8 . = 35.3U8 cu. ft. = 1058 X io 1 ergs. I litre = 1000 cm. 3 i k. degree C. = 4200 joules. = 1.7598 pints. = 4200 X io 7 ergs. = .22 imp. gal. i calorie = i k. raised i C. i imp. gal. = .1605 cu. ft. = 426.9 km. = 277.27 cu. ins. = 3080.9 ft. -Ibs. = 4.545963 litres. i watt = i joule per sec. i U. S. gal. = 231 cu. ins. f work done by f ,. = .83254 imp. gal. ( 981 cm. per ~ ( sec. per sec. J a current of 1 i amp. at i [ volt. _ j 32.2 ft. per } i horse-power = 550 Ibs. per sec. sec. per sec. ( 746 X io 9 ergs g at Greenwich = 32.19078 ft. = ! } per sec. = 981.17 cm. = 746 watts. g at London = 32.182 ft. _ j i.oi forcesrde- = 980 9 cm. ~ \ cheval. g at Manchester = 32.196 ft. = 981.34 cm. iforce-de-cheval= (-9863 horse- ( power. g at the equator = 32.088 ft. = 736 watts. g at Baltimore =J978.O4 cm. = 32.152 ft. _ j 545 ft. -Ibs. per ~~ / sec. = 980 cm. = 75 km. per sec. g at Montreal = 32.1765 ft. i radian =57.296 degrees. = 980.73 cm. To convert common into hyper- The inertia or mass of a body f its wt. in Ibs. I = \ at London ' [ -4-32.2. bolic and hyperbolic into common logarithms, multiply the former by 2.3025 and the latter by .43429. HYDRAULICS, CHAPTER I. GENERAL PRINCIPLES. FLOW THROUGH ORIFICES, OVER; WEIRS, ETC. I. Fluid Motion. The term " hydraulics," as its deriva- tion (vdoop, water; <*uAo.c, a tube or pipe) indicates, was primarily applied to the conveyance of water in a tube or pipe, but its meaning now embraces the experimental theory of the motion of fluids. The motion of a fluid is said to be steady or permanent when the molecules successively arriving at any given point are animated with the same velocity, are subjected to the same pressure, and are the same in density. As soon as the motion of a stream becomes steady a permanent regime is said to be established, and hydraulic investigations are usually made on the hypothesis of a permanent regime. With such an hypothe- sis, any portion of the fluid mass, which leaves a given region, is replaced by a like portion under conditions which are identi- cally the same. The terms "steady motion" and "permanent regime" are often considered to be synonymous. a FLUID MOTION. The general problem of flow is the determination of the relation which exists at any point between the density, pres- sure, and velocity of the molecules which successively pass that point. The actual motion of a fluid is exceedingly complex, and, In order to simplify the investigations, various assumptions are made as to the nature of the flow. 2. (a) Stream-line Motion. The molecules may be re- garded as flowing along definite paths, and a succession of such molecules as forming a continuous fluid rope, which is termed an elementary stream or a fluid filament ; or, if the motion is steady, and the paths therefore fixed in space, is termed a stream-line. Experiment shows that the velocity of flow in any cross- section varies from point to point, and it is often assumed that the section is made up of an infinite number of indefinitely small areas, each area being the section of a fluid filament. (b) Motion in Plane Layers. In this motion it is assumed that the molecules, which at any given moment are found in a plane layer, will remain in a plane layer after they have moved into any new position. (c) Laminar Motion. On this hypothesis the stream is supposed to consist of an infinite number of indefinitely thin layers. The variation in velocity from point to point of a cross-section may then be allowed for, by giving the several layers different velocities based upon the law of fluid resistance between consecutive layers. 3. Density; Compressibility; Head; Continuity. The freezing-point of pure water is 32 F. or o C. " boiling- " " " " " 212 F. or 100 C. " max. density " " " 44 39. i F. or 4 C. -" standard mean temperature " 62 F. or i6.66 C. The comparative densities and also the comparative vol- umes are the same at 32 F. and 46 F. FLUID DENSITY. 3 The bulk of fresh snow is 12 times the bulk of the equiva- lent water. i cu. ft. of fresh snow weighs 5.2 Ibs. and its s. g. is .0833. I cu. it. of ice at 32 F. weighs 57 J Ibs. and its s. g. is .922. i cu. ft. of average sea-water at 62 F. weighs 64 Ibs. and Its average s. g. is 1.028. i cu. ft. of pure water at 32 F. weighs 62.418 Ibs. *' " 39. i F. " 62.425 " 44 44 44 52. 3 F. " 62.400 " 44 44 44 44 5 2 p 44 62. 355 *' 44 44 2I2 p 44 59.640 " 6.2355 gallons or contains , ,, . 6.2328 imperial gallons. i cu. yd. " " " 168.36 gallons. i cu. metre " " " 220.09 " The vol. of i Ib. of pure water at 32 F. ' is .016021 cu. ft. " " l< 39. i F. " .016019 i4 52. 3 F. <4 .016 62 F. " .016037 " j 2 p t< .01677 '* The vol. of i ton " " 52. 3 F. "35.9 <4 44 4< 4t sea-water at 62 F. "35 " i tonne of pure water at 39. i F. "35.3156 " -0353 4< 44 4 ' i gallon of pure water at 62 F. weighs 10 Ibs. and its vol. = 277.123 cu. ins. = .16037 cu - ft- i imperial gallon of pure water at 62 F. weighs 10.00545 Ibs. and its vol. = 277.274 cu. ins. FLUID DENSITY. |P ! ! i & III I |J CO "1 M M -.a CO M ^' < MOO O M 11 M M M M ' MMM O t^ OOO O vn nparative Density. O CO O -1- M M OO M r>i O !-> u-> co O co co r- r^ r^- M Q h^ d U") ON 1O co m o* O \O c^ O vO O *O w~i tr> 10 10 ** ''I- rfr CO CO MMM o U -T3 . vn co co 0.2 Jj M CO O Tf -^b/D M M C4 CO CO Tf IT) o o r^co o o o M w M CO *^" u^ \O f^* 1^ CO O O u~> CO O CO ^3 MM d|| OO O O **? M CO O ~f M CO O ~t M CO O Tj- M O M co 10 O co O O M *? O O M O ta .S c ' 5 ? MM co M r^ O O co M CO CO O M 4-1 [A O O O O O O O 88 8 8 si- O ' O O O O O MM M O O O O M M M M ll co M \r> \r> M CO O MM O 8-1- M vn -J- O !~ ""> 'I' -'.5 "**<$ **** *t CO CO CO CO c 3 MM MM MM M MM MM M 'i 00 00 O O O O O O O O ^"o u oo r- co M O -f-TfMCO M ^--l- *^-M 'w ^ O M OOO oo M . r^ o M co M-or^-M r^ OM coco ct ~ CO O OOO ooo co co r^o in -T N M O COO ~f M II O O O O- O O O OOO O-OOO co coco coco O M w U ilS S 5 ^ O .r- rc oo vn O co M r^ u-> o co I'lftx- o M M M co TJ- ;^ u~) \o i^. r> r^ co o O M M.M M coTj-i/->u->voo rococo OO U r 'C co O ^t" *~^ - CO O -^- M CO tOO "3- M CO O ~t M |ji & Cl ro 10 u-> r O O M N T^- U-.O O CO O M M covouor^OO O M M -txnoco [2 FLUID COMPRESSIBILITY. 5 The temperatures in this table may be taken as abscissae, and trm corresponding values in the three remaining columns as ordinates. Curves of comparative density, weight per cubic oot, and weight per gallon are thus obtained, and the values orresponding to any specified temperature can be easily and /ery accurately determined from these curves by direct meas- urement. The weights per cubic foot in the table have been calcu- lated by means of Rankine's approximate formula, w 1000 T 62.425 " T* -(- 250,000 ' w being the weight per cubic foot corresponding to the -absolute temperature T, i.e., 461 -)- ordinary temperature. The specific weights obtained by this rule for the lower temperatures are very exact, but for the higher temperatures they become too large. Thus the rule gives 59.76 Ibs. as the weight of a cubic foot of pure water at 212 F., while actual measurement makes the weight 59.64 Ibs. The comparative densities between o C. and 40 C. are the values obtained by Chappuis. Compressibility. Fluids are sensibly compressible under heavy pressures, and the compression is proportional to the pressure up to about 1000 Ibs. (68 atmospheres) per sq. inch. Grassi's experiments indicate that the compressibility of water diminishes as the temperature increases. Water compresses about 47i millionths (i.e., ^^ = ^^, nearly) of its bulk for each atmosphere. This is equivalent, approxi- mately, to a reduction of y 1 ^ in the bulk under a pressure of 2 tons per sq. inch. If a volume F of a fluid is compressed by an amount A V under an increase Ap of the pressure, then the amouut of com- pression per unit of vol. is AV -FT- and is called the cubical compression. The ratio of the FLUID COMPRESSIBILITY. TABLE OF ELASTICITY OF VOLUME OF LIQUIDS. (Reduced from Grassi's results.) Liquid. Elasticity oi Voiume. Temperalure. Mercury . .. Water . 717,000,000 j 42,000,000 o C. o C. Sea- water . . Ether Alcohol.... Oil } 45,900,000 52,900,000 ( 16,280,000 I 15,000,000 j 25.470,000 I 23,380,000 44,000 ooo 18 C. o C. 14 C. 7-3C. 13.1 C. N. B. The value for mercury is probably erroneous. increment of pressure to the cubical expansion, viz., is termed the elasticity of volume. This is sensibly constant within wide limits, and is generally denoted by the letters Dor K. The vertical distance between the free surface of a mass of water and any datum plane is called the head with respect to that plane. If the water extends down to the level of the plane, a pressure p is produced at that level, and the value of /, so long as the water is at rest, is given by the equation W W w being the specific weight of the water and p Q the pressure at the free surface. Thus the pressure may be measured in terms of the head, and hence the expression " head due to pressure " or " pressure head." A column of water at 62 F. and 2.3093 ft. in height exerts a pressure of I Ib. per sq. inch. A column of water at 62 F. and 33.947 ft. or 10.347 metres in height exerts a pressure of 14.7 Ibs. per sq. in., or one atmosphere. A column of water at 62 F. and I ft. in height exerts a. pressure of .433 Ibs. per sq. inch. HEAD OF WATER. ^ Head. A head of water is a source of energy. A volume o water descending from an upper to a lower level may be employed to drive a machine, which receives energy from the water and again utilizes it in overcoming the resistances o other machines doing useful work. Let Q cu. ft. of water per second fall through a vertica. distance of // ft. Then the total power of the fall = wQlt w Qh ft.-lbs., = - h.p. , w being the weight of the water in pounds- per cubic foot. Let K be the proportion of the total power which is. absorbed in overcoming frictional and other resistances. Then the effective power of the fall = wQ/i(i K), and the effi- ciency is I K. Continuity. Imagine abounding surface enclosing a space of invariable volume in the midst of a moving mass of fluid. The principle of continuity affirms that, in any interval of time, the flow into the space must be equal to the outflow during the same interval. Giving the inflow a positive and the outflow a negative sign, the principle may be expressed symbolically by The continuity of a mass of water will be preserved so long as the pressure exceeds the tension of the air held in solution. It is on account of the pressure of this air that pumps cannot draw water to the full height of the water-barometer, or about 34ft. Generally speaking, the pressure at every point of a con- tinuous fluid must be positive. A negative pressure is equiva- lent to a tension which will tend to break up the continuity presupposed by the formulae. Should negative pressures result from the calculations, the inference would be that the latter are based upon insufficient hypotheses. The pressure in water flowing through the air cannot at any point fall below the atmospheric pressure. There are cases,, 8 BERNOULLI'S THEOREM. however, as when water flows through a closed pipe (Art. 6, Chap. II), in which the pressure may fall below this limit and become almost nil. In this case there is a danger of the air held in solution being set free, thus tending to interrupt the continuity of the flow, which may even be wholly stopped if the air is present in sufficient volume. Consider a length of a canal or stream bounded by two normal sections of areas A lt A 2 , and let 7',, ?>., be the mean normal velocities of flow across these sections. Then, by the principle of continuity, and the velocities are inversely as the sectional areas. Again, assume that a moving mass of fluid consists of an infinite number of stream-lines, and consider a portion of the mass bounded by stream -lines and by two planes of areas A x , A 2 at right angles to the direction of flow. If i\ , ?'., are the mean velocities of flow across the planes, v l A l = Q = i' 2 A if the fluid is incompressible. Assuming that the fluid is compressible, and that the mean specific weights at the two planes are w l and w., , then the weight of fluid flowing across A. Y is equal to the weight which flows across A 2 , since the weight of fluid between the two planes remains constant. Hence 4. Bernouilli's Theorem. This theorem is based on the following assumptions: (1) That the fluid mass under consideration is a steadily moving stream made up of an infinite number of stream-lines whose paths in space are necessarily fixed. (2) That the velocities of consecutive stream-lines are not widely different, so that viscosity, or the frictional resistance between the stream-lines, is sufficiently small to be disregarded. BERNOULLI'S THEOREM. (3) That the fluid is incompressible, so that there can be no. internal work due to a change of volume. In any given stream-line let a portion AB, Fig. I, of the fluid move into the position A'B in t seconds. FIG. i Let a l , pi , i\ , z l be the normal sectional area, the intensity of the pressure, the velocity of flow, and the elevation above a datum plane zz of the fluid at A. Let a 2 , / 2 , z/ 2 , # 2 denote similar quantities at B. Since the internal work is nil, the work done by external forces must be equivalent to the change of kinetic energy. ( the work done by gravity ( -j- the work done by pressure. But when the fluid AB passes into the position A'B', the work done by gravity is equivalent to the work done in the transference of the portion BB' ', and therefore, t being the time, the work done by gravity = wa l . AA'. z l Z/# 2 . BB' . z z ~ wa . it . wa . vt . z Now the external work = since A A' = it, BB' = and \ = Q = a 2 v r Again, the ivork done by the pressures on the ends A and B The work done by the pressure on the surface of the io BERNOUILLrS THEOREM. stream-line between A and B is nil, since the pressure is at every point normal to the direction of motion, The change of kinetic energy = kinetic energy of A' B' kinetic energy of AB = kinetic energy of BB' kinetic energy of A A', since the motion is steady, and there is therefore no change in the kinetic energy of the intermediate portion A' B. Thus the change of kinetic energy = - aJBB' --- a^AA'- W 7> 2 W V, 2 * w ^ (v? 7A 2 \ = -Ot I- 2 - -^- . g " \ 2 2 / Hence, equating the external work and the change of kinetic energy, which may be written in the form w v* w v 2 - = ^ 2 + A+-. (i). or But ^4 and .# are arbitrarily chosen points, and therefore, at any point of a stream-line, the motion being steady and the viscosity nil, the gradual interchange of the energies, due to head, pressure, and velocity, is expressed by the equation W V 2 wz + P H --- L = W H, a constant ; . . . (3) o or Z + w + ^ = H> a Constant 5 ... (4) Z being the elevation of the point above the datum plane, p the pressure at the point, w the specific weight, and v the velocity of flow. This is Bernouilli's theorem. BERNOULLI'S THEOREM. II Thus the total constant energy of wH ft.-lbs. per cubic foot of fluid, or H ft.-lbs. per pound of fluid, is distributed uniformly along a stream-line, wH being made up of wz ft.-lbs. due ta head, p ft.-lbs. due to pressure, -- ft.-lbs. due to velocity, p and H being made up of z ft.-lbs. due to head, ft.-lbs. due w V 2 to pressure, and ft.-lbs. due to velocity. Hence the total energy is made up of three elements, and each element may be utilized by a specially designed motor. The now almost obsolete overshot-wheel is driven by the weight of the water filling the buckets on one side and descend- ing from a higher to a lower level. In the breast-wheel and certain turbines, the energies, due both to the weight (wz) and lwv\ to the velocity ( - 1, are transformed into useful work. The / wv* \ rotation of impulse-wheels is due to the kinetic energy ( - j : of a jet of water issuing from a nozzle and impinging upon curved buckets. Finally, the piston of the hydraulic engine is. actuated by water admitted into the cylinder from a closed pipe in which the water under pressure moves with a low* velocity. Assuming that (a) the motion is steady, (<) the frictional resistance may be disregarded, (c) the fluid is incompressible, Bernoulli's theorem may be applied to currents of finite size at any normal section, if the stream-lines across that section are sensibly rectilinear and parallel. Tr^ere is then no interior work due to a change of volume, and the distribution of the pressure in the section under consideration will be the same as if the fluid were at rest, that is, in accordance with the hydro- static law. This is also true whether the flow takes place 12 APPLICATIONS OF BERNOULLI'S THEOREM. under atmospheric pressure only, or whether the fluid is wholly or partially confined by solid boundaries, as in pipes and canals, or whether the flow is through another medium already occupied by a volume of the fluid at rest or moving steadily in a parallel direction. In the last case there must necessarily be a lateral connection between the two fluids, but the pressure over the section must follow the hydrostatic law throughout the separate fluids, and there can be no sudden change of pressure at the surface of separation, as this would lead to an interrup- tion of the continuity. The hypotheses, however, upon which these results are based, are never exactly realized in actual experience, and the results can only be regarded as tentative. Further, they can only apply to an indefinitely short length of the current, as the viscosity, which is proportional to the surface of contact, would otherwise become too great to be disregarded. 5. Applications of Bernouilli's Theorem. If a glass tube, open at both ends, and called a piezometer (nie^eiv, to press; jjerpov, a measure) is inserted vertically in the current, Fig. 2, at a point N, z ft. above the point in the datum line, the water will rise in the tube to a height MN dependent upon the pressure at N. The effect of the eddy motion, produced at TV by obstruct- i ing the stream-lines, may be dimin- ished by making this end of the _ j, _ tube parallel to the direction of flow. Disregarding the effect of the eddies and taking / and / to be the intensities of the pressure at N and of the atmospheric pressure, respectively, then, w w APPLICATIONS OF BERNOULLI'S THEOREM. and therefore - w tv = ON -\-MN-\- ~ w The locus of all such points as J/is often designated "the line of hydraulic gradient," or the "virtual slope," terms also used when friction is taken into account. Let the two piezometers AB, CD, Fig. 3, be inserted in the current at any two points B and D, z^ ft. and 3 2 ft. respectively above the points E and F'm the datum line. FIG. 3. / l be the intensity of the pressure at B in pounds per square foot, / 2 that at D, and let the water rise in these tubes to the heights BA, DC. Then, A. w ,' and 14 APPLICATIONS OF BERNOULLI'S THEOREM. and therefore (6) the line AG being parallel to the datum line. Thus, (z. + - 1 ) (V, + - 2 ) is equal to the fall of the free \ 1 ' wl \ 2 iv I surface level between the points B and D. Let v^ , 7> 2 be the velocities of floW at B and D. Then, by Bernoulli's theorem, and therefore the fall of free surface level between B and D Equation (7) may also be written in the form + ~) - (* + ~) = ^ + CG * ' ( 8 ) so that the velocity at D is equal to that acquired by a.body with an initial velocity i\ falling- freely through the vertical distance CG. Froude illustrated Bernoulli's theorem experimentally by means of a tube of varying section, Fig. 4, conveying a current between two cisterns. The pressure at different points along the tube is measured by piezometers, and it is found that the water stands higher and the pressure is therefore greater, where the cross-section is larger and the velocity consequently less. Reynolds illustrates the principle, that the pressure in a frictionless pipe of varying section increases and diminishes with the section, by forcing water at a high velocity through a :J-in. pipe drawn down in the middle to a bore of .05 inch. At APPLICATIONS OF BERNOULLI'S THEOREM. 15 this point the pressure is so much diminished, that the water hisses and boils. To the same cause is due the hissing sound heard in water-injectors and in partially opened valves. If the section of the throat at A is such, that the velocity is that acquired by a body falling freely through the vertical distance FIG. 4. h between A and the surface level of the water in the cistern, and if p be the pressure at A , and z the elevation of A above datum, then, neglecting friction, p 7> 2 P z + + - = H = s + // + /0 . 1 ' W 2g W But ?' 2 = 2g/i, and therefore p = / , so that the pressure at A is that due to atmospheric pressure only. Thus, a portion of the pipe in the neighborhood of A may be removed, as in the throat of the injector. Again, let the cross-section in the throat at B be less than that at A. The pressure at B will be less than the atmos- pheric pressure, and a column of water will be lifted up in the urved piezometer to a height //'. Let a^ , s lt /! , i\ be the sectional area, elevation above datum, pressure, and velocity at B. Let a 2 , # 2 , / 2 , 7' 2 be similar symbols at E. 16 APPLICATIONS OF BERNOUILU'S THEOREM. Then P Put H 2 = ^ 2 -f- ~ > the height above datum to which the water is observed to rise in the piezometer inserted at t and also let H. = s. + ~ - //'. Then 1 zc; * i 2g 2 g a. since a. 2 v 2 = a^\ , , being the sectional area at E. Therefore an equation giving the theoretical velocity of flow at the throat B. Hence the theoretical quantity of flow across the section at B is \'a* - a* V2g(H 2 - //J. . . (10) This is the principle of the aspirator and also of the Venturi water-meter, which, as now used, is said to be correct to within per cent. The actual quantity of flow is found by multiplying equa- tion (10) by a coefficient C, whose value is to be determined by experiment and may be taken to be approximately unity. If the pressure at E is positive, then H 2 is merely the height to which the water is observed to rise in an ordinary piezometer inserted at E. Again, Froude also points out that when any number of combinations of enlargements and contractions occur in a pipe, the pressures on the converging and diverging portions of the EXAMPLES. if pipe will balance each other if the sectional areas and direc- tions* of the ends are the same. Ex. i. One cubic foot of water per second flows steadily through a frictionless pipe. At a point A, 100 ft. above datum, the sectional area of the pipe is .125 sq. ft., and the pressure is 2500 Ibs. per sq. ft. Find the total energy at A per cubic foot of water. At a point B in the datum line, the pressure is 1250 Ibs. per sq. ft. and the sectional area .0625 3q. ft. Find the loss of energy per cubic foot of water between A and B. The velocity of flow at A = - = 8 ft. per sec. The total energy at A per cubic foot of water The velocity of flow at B = > = 16 ft. per sec. .0025 The total energy at B per cubic foot of water 1250 i6 2 = + -65T + -64- == 24 ft - lbs ' Hence, the loss of energy between A and B per cubic foot of water = 141 24 = 117 ft. -Ibs. Ex. 2. A horizontal frictionless pipe, in which the pressure is 100 Ibs. per square inch, gradually contracts to a throat of one tenth of the diameter and then again gradually enlarges to a pipe of uniform diame- ter. What will be the maximum velocity of flow at the throat ? The velocity at the throat will be greatest when the pressure there is nil. Hence, if v is the throat velocity and therefore the pipe velocity, TOO x 144 i / v \*_ v 1 62^ * 64\mo/ ~ f 64' and v = 121.437 ft. per sec. 6. Rotation of a Fluid. In any stream-line moving freely in space, let A BCD be an element of mass m and normal thickness dn( = BC). It is acted upon by the pressures on AD and BC, a pressure of intensity p on the area AB(= a), a pressure of intensity / -f- dp on the area CD, its weight i8 ROTATION OF FLUIDS. inclined at an angle a to the normal, and the centrifugal force m , r being the radius of curvature. . ' i > -0,. Resolving along the normal, or or a . dp ;;/ mg cos a = o, /z> 2 \ iva . dn tv 2 \ a. dp-- m{ + g cos aj = - { + g cos orj, dp wlv* \ --7- = I r- g cos orJ. dn g \ r I If the stream-line is in a horizontal plane, a 90, and then, dp _ w v 2 dn cf r But by equation (4), Art. 4, since z is now constant, dH ^_ dp v dv_Wv dv\_2v ffv dv\ dn ~~ w " dn ' g " dn ~ g\r ~* dn^ " " g * 2 T """ dn/ * WHIRLING FLUIDS. l& i fv dv\ /The expression I- h T~J * s designated the average angular velocity, or the rotation of the fluid. Again, if the stream-line is horizontal and is also circular,., dn dr, and dp wv 2 dr ~~ g "r " ' a differential equation connecting the pressure and the velocity.. If v is a known function of r, the pressure can be at once determined. 7. Whirling Fluids. Let a fluid mass whirl like a rigid body about a vertical axis YY, with an angular velocity GO. Consider the relative equilibrium of an element of mass im v ^ v FIG. 6. at P distant x horizontally from the axis and y vertically from- the origin O in YY. Take PA horizontally to represent the centrifugal force maPx, and PB vertically to represent the weight mg. The remaining forces must be equal and opposite to the resultant of these two forces, viz., the diagonal PC of the parallelogram AB. The magnitude of this resultant is PC = 20 LEVEL SURFACES. Its slope, a, is given by Integrating, dy mg -j- =. tan a = 5 dx = .__ moj^x ' GO*X' c being a constant of integration. Thus an infinite number of logarithmic curves can be drawn such that the tangent at any point in any one of the curves is in the direction of the resultant force at that point. These curves are called lines of force, and the surfaces cutting these lines of force orthogonally are designated level or equipotential surfaces. If ft is the slope of a level surface, then dy 4- -~ tan 6 cot a = = dx *" ^ mg Integrating, c being a constant of integration. Thus the level surfaces are paraboloids of revolution. For the free surface this result is obtained more simply as follows : The fluid element of mass m in the free surface at P is kept in relative equilibrium by (a) the centrifugal force m&Px ()its weight mg, and (c] the fluid pressures, which must neces EXAMPLES. 21 sarily have a resultant normal to the free surface at P. Draw- ing- j:he horizontal PN and the normal PG to meet the axis of rotation in N and G, PNG is evidently a triangle of forces,. NG mg NG g and therefore - 7T1 r 7 = =- = , and NG = :., a constant. PN mz&x x GO* Thus, the sub-normal is constant, and the free surface must be a paraboloid with its vertex at the point O where the free surface cuts the axis of rotation. Ex. i. Deduce the law of pressure variation (a) for water in a vessel moving slowly towards a hole in the centre, the stream-lines being ap- proximately horizontal circles and the velocity of any fluid particle inversely as its distance from the axis (b) for water rotating as a rigid body about an axis (as in a full centrifugal pump before delivery com- mences), the velocity of any fluid particle being directly proportional to- its distance from the axis. (a) Take v = , then i dp i if_ _ a^ vj dr ~ g r ~ g r 3 ' Therefore = c ^- = ^ - . TV 2F r 2P r (b) Take v = br, then *x - "'} ' \ dp i v* i - -r- = = b*r. iv dr g r g 6 i z/ a Therefore . = *>+ r a =<:'+. W 2g 2g Ex. 2. A cylindrical vessel, 10 ft. in height and i ft. in diam., is half full of water. Find the number of revolutions per minute which the vessel must make so that the water may just reach the top, the axis of revolution being coincident with (a) the axis of the vessel, (b) a gen- erating line. (a) The free surface of the water is the paraboloid POP, Fig. 8, with its vertex at O, since the vol. of the paraboloid = vol. of circumscribing cylinder, = vol. of water in vessel. g latus rectum I PN* I Then &~ NG = = ^ = = 5-, co s 22 ON 80 and GO = 4/32 x 80 = 161/10. SHARP-EDGED ORIFICES. The linear speed of the rim at P = J &? = 8|/io, .and the number of revols. per min. = - H = 482.96. -y- x i N FIG. 8. FIG. 9. (6) The free surface is now the paraboloid OP, with its vertex at 6>, Fig. 9. g i /W a i ^ = 2" C?A r ~ ' Then .and a? = 4/640 = Thus the number of revols. per min. = 20 8. Orifice in a Thin Plate. If an opening is made in the "wall or bottom of a tank containing water, the fluid particles immediately move towards the opening, and arrive there with a velocity depending upon its depth below the free surface. The opening is termed an " orifice in a thin plate," when the "water- springs clear from the inner edge, and escapes without .again touching the sides of the orifice. This occurs when the bounding surface is changed to a sharp edge, as in Fig. 10, .and also when the ratio of the thickness of the bounding sur- face to the least transverse dimension of the orifice, does not SHARP-EDGED ORIFICES. 50. FIG. 13. Thus the areas of the contracted section and of the orifice are in the ratio of 16 to 25, and, generally speaking, this is assumed to be the ratio whatever may be the form of the orifice. TORRICELLI'S THEOREM. 9. Torricelli's Theorem. Let Fig. 14 represent a jet issuing from a thin-plate orifice in the side of a vessel contain- ing water kept at a constant level AB Let XX be the datum line, MN the contracted section, and consider any stream-line mn y m being in a region where the FIG. 14. velocity is sensibly zero, and n in the contracted section. Then, by Bernouilli's theorem, the motion being steady, A P ? i + + = 2 + -f ' W 2g ""' " (0 /, / x being the pressures at w and m, and #, ^. their elevations above datum. Hence - = z, z -4- 2g If the flow is into the atmosphere, ' p = the atmospheric pressure A = w p p (2V , and O being the point in which the vertical through m intersects. the free surface. Thus . (3) li being" the depth of n below the free surface. TORRICELLI'S THEOREM. The result given by equation (3) was first deduced by Totricelli. The depth below the free surface is very nearly the same for .all points of the contracted vein, and the value of v as given by (3) is taken to be the theoretical mean velocity of flow across the contracted section. Equation (3) is equivalent to the statement that when the orifice is opened, the hydrostatic energy of the water, viz., li ft.-lbs. per pound, is converted into the kinetic energy of V 2 ft.-lbs. per pound. Thus, if the jet is directed vertically upwards, it will very nearly rise to the level of the free surface, and would reach that level were it not for air resistance, or for viscosity, or for friction against the sides of the orifice, or for a combination of these retarding causes. If the jet issues in any other direction, it describes a para- bolic arc of which the directrix lies in the free surface. Let OTV> Fig. 15, be such a jet, its direction at the orifice at O making an angle a with the vertical. With a properly FIG. 15. formed orifice a greater or less length of the jet will have the appearance of a glass rod, and if this portion were suddenly solidified and supported at the ends, it would stand as an arch without any shearing stress in normal sections 26 VESSELS IN MOTION. Again, the horizontal component of the velocity of flow at any point of the jet is constant (= v sin -), so that, for the unbroken portion of the jet, equidistant vertical planes will intercept equal amounts. of water, and the height of the C. G. of the jet above the horizontal line OV, will be two thirds of the height of the jet. 10. Efflux through an Orifice in the Bottom or in the Side of a Vessel in Motion. If a vessel containing water z ft. deep ascend or descend vertically with an acceleration /, the pressure / at the bottom is given by the equation being the atmospheric pressure. Therefore q Before an orifice is opened, if the heavier vessel is reduced to rest by applying an upward acceleration f, the pressure at the depth z is changed from ws to wz\l -\- j, O ' while in the other vessel it would be changed from ws to wzll -j. If now an orifice is opened at the bottom, the velocity of efflux v is still taken as being due to the head of the pressure /, and therefore by Torncelli's Theorem FIG. 16. = 21 - f\ . ' Let W l be the weight of the vessel and water, and let the vessel be connected with a counterpoise of weight W 2 by FLOW IN FRICTION LESS PIPE. 27 means of a rope passing over a pulley. Then by Newton's second law of motion, and neglecting pulley friction, f T W, T being the tension of the rope. Therefore, also, T= l Next let a cylindrical vessel, Fig. 17, of radius r and containing water, rotate with an angular ve- locity oo about its axis, Art. 7. The surface of the water assumes the form of a paraboloid with its vertex at O and its latus rectum equal 2g to -. If an orifice is made at Q CtT in the side of the vessel, at a verti- FIG. 17. cal distance z from O, the water will flow out with a velocity 7' due to the head of pressure at the orifice. This head is PQ, and z. the sign being plus or minus , according as the orifice is below or above O. Hence, by Torricelli's theorem, or V 2 = 2gZ. ii. Application to the Flow through a Frictionless Pipe t)f Gradually Changing Section (Fig. 18). Let the pipe be supplied from a mass of water of which the free surface is H ft. above datum. 28 FLOW IN FRICTION LESS PIPE. Let 0j , /! , ^ be the sectional area, pressure, and velocity- of flow at any point A , ^ ft. above datum and /*j ft. below the free surface. Let # 2 , / 2 , v 2 be similar symbols for a second point B> z^ ft. above datum and /i 2 ft. below the free surface. FIG. 18. Then by the condition of continuity, and by Torricelli's theorem, A -A and *, 2 Hence o tr A w HYDRAULIC COEFFICIENTS. 29 so that Bernouilli's theorem, viz., 2 at the vena contracta is a little less than \/2gh, the theoretical velocity (Art. 9), and the ratio of v to ^2gh is called the coefficient of velocity. Denoting this coefficient by c v , then, COEFFICIENT OF VELOCITY. and the equations for the velocities of discharge in the case of moving vessels (Art. 10) become and it = v. 2(g f)/i ' V* = C v \uW 2gs). A mean value of c v for- well formed simple orifices is .974. Assuming that the face of the orifice is vertical and that the jet issues horizontally with a velocity of v ft. per second, under a head of // ft. of water, and assuming also that in t sees., a FIG. 20. fluid particle reaches a point y ft. measured vertically and x ft. measured horizontally from the point of discharge, then, dis- regarding the effect of air resistance and other disturbing causes, x = y = and therefore X* 2V* 2 - = - - = C*2gh y g g v * or s- P.. - 32 COEFFICIENT Or VELOCITY. If x v , jj/ 1 are the coordinates of the fluid particle in any other position, then, also, Hence x 2 - x a i -- y)' which is the formula used in the McGill laboratory in the experimental determination of coefficients of velocity. The position of the jet is defined by vertical measurements from a straight-edge, supported horizontally above the jet, by a bracket on the tank face at one end, and at the other on a bearing, which admits of a vertical adjustment, Fig. 2 1 . FIG. 21. The straight-edge is of machinery-steel, is 5^ ft. in length, 2j ins. in depth, f in. in width, and is graduated so as to give the horizontal distances from the inner face of the orifice' plate. The vertical ordinates are measured by a Vernier caliper specially adapted for the purpose. The flat face of the movable limb is made to rest upon the upper surface of the straight- edge, and the caliper-arm hangs vertically. A bent piece of \vire, with a needle-point, is clamped to the other limb, and, COEFFICIENT OF VELOCITY. 33 by means of the screw adjustment, can be readily moved until it jitst touches the upper or lower surface of the jet. By means of the above method, an extended series of 'experiments with i-in., ^-in., and i-in. sharp-edge orifices, and sunder heads varying from 6 to 20 ft., gave .99 as the average ^value of the coefficient of velocity (c v ). Let the direction of the jet, Fig. 22, at the point of dis- charge make an angle a with the horizontal, and let^/j^, FIG. 22. -*-2i y*i b e the coordinates defining the position of a fluid particle after intervals of t l sees, and t 2 sees. Then jtr l = i' cos a , / t and jj', = i' sin a . / t ^ = f cos . /.j and y^ = v sin a . / 2 These equations give tan *!**(** ~ ^l and * sec 2 a 2v* 2 ' sec 2 a l tan - J t g x. 2 tan a yj from which a: and then c n can be calculated. 34 COEFFICIENTS OF RESISTANCE AND CONTRACTION. (b) Coefficient of Resistance. Let h v be the head required to produce the velocity v. Let h r be the head required to> overcome the frictional resistance. Then //, the total head, = h v + h r h v (i + . Can. Soc. C. E., 1900). The miner's inch of water is an arbitrary module adopted in mining districts for selling water. It is variously defined as being the amount of water discharged per minute by an orifice I in. square, or an equivalent fraction of a larger orifice, with a head of from 6 to 9 ins., the thickness of the orifice being usually 2 inches. FIG. 30. One great difficulty is that this is a variable quantity de- pending upon the specified head, and therefore all such mod- ules should also define the flow in cubic feet per minute. There are many practical difficulties in the way of deliver- ing absolutely exact quantities of water, but the definition of the module or unit should be correct within a reasonable limit of error. If it is a definition of a single miner's inch from an orifice of i sq. in., it should go no further; but if the inch is defined as being some fractional part of the discharge from a. MINER'S INCH. 45 larger orifice, it should be limited to the capacity of that orifice. Fturther, as it is a term of local signification only, the dis- charge should be given in cubic feet per minute, convenient discharges being i| and 2 cu. ft. The flow under low heads is irregular. Heads of I ft. or more are not suitable, because o the water is delivered from ditches or flumes in which the depth is never great. The question thus resolves itself into a choice of a stand- ard module or unit from a flow under one of two conditions, viz. : (1) With a low head of 6J ins. above the centre of the orifice giving a discharge of i cu. ft. per minute, with the advantage that it is already practically recognized as the miner's inch, and with the disadvantage that the flow is irreg- ular. (2) With a head of 1 1 J ins. above the centre of the orifice, and a discharge of 2 cu. ft. per minute, the flow being much more regular, but the quantity discharged is not recognized in practice. The flow under the first condition is chosen as being the one now in use in British Columbia, and the following specifica- tion is given of the miner's inch, including discharges of from I to 100 miner's inches of i cu. ft. per minute: The water taken into a ditch or sluice shall be measured at the ditch or sluice head. It shall be taken from the main ditch, flume, or canal, through a box or reservoir arranged at the side, and the water shall have no appreciable velocity of approach. The orifice shall be fixed vertically at right angles to the delivering waterway, and the edges and corners shall be square and sharp, the top, bottom, and sides of the orifice being at right angles to the pressure-board. The issuing vein shall be fully contracted, and the discharge shall pass freely into the air. The distance between the sides and bottom of the orifice and the sides and bottom of the waterway shall be at least three (3) times the least dimension of the orifice. The miner's inch of water shall mean T l of the quantity which shall 4 6 MINER'S INCH. be discharged through an orifice six (6) ins. wide and two (2) ins. high, made of 2-in. planks, planed, made smooth and painted. The water shall have a constant head of 6J ins. above the centre of the orifice, and the amount discharged shall be estimated at ij cu. ft. per minute. Discharges up to and including 101.55 miner's inches of I J cu. ft. of water per minute shall be as in the following table * Dimensions of Orifice in Inches. Head in Inches Numberof Miner's /^- * f Tnrhfs nf i f^iihin Width. Depth. Orifice. Feet per Minute. 6 2 6.25 11.9858 12 2 6.25 24.2485 18 2 6.25 36-3851 24 2 6.25 48.6865 4 4 6.25 15-6998 6 4 6.25 23.5560 12 4 6.25 47.2853 18 4 6.25 71 .6296 25i 4 6.2 5 101.5495 T. Drummond, B.A.Sc., has made an interesting series of experiments (Trans. Can. Soc. C. E., vol. XIV, 1900) on the Miner's Inch, in the Hydraulic Laboratory, McGill University. The discharges recorded were made under low heads of from 6 to 12 ins., and with two kinds of orifices, viz. : (1) Standard sharp-edged rectangular orifices in brass from 1 to 4 sq. ins. in area. (2) Square-edged rectangular orifices in wood, 2 ins. thick,, 2 to 4 ins. in height, and J- to 24 ins. in width. The formula adopted for the discharge was Q = \CB tiTg(Hf - H*) (see Article 22), in which C is the coefficient of discharge ; g\* 32.176; Q is the discharge in cubic feet per second ; B is the width of the orifice ; HI and H 2 the heads over the top and bottom of the orifice. No corrections were made for changes in temperature. MINER'S INCH. 47 The shape of the orifice has a very sensible effect upon the discharge. Circular orifices give the least discharge, the greatest discharges occur with rectangular orifices, while the discharges with square orifices are intermediate. The coefficient of discharge (C) diminishes as the size of the orifice increases, the same form of orifice being maintained. For the same orifice C diminishes as the head increases. In rectangular orifices of constant depth the coefficient of discharge increases with the width. If the width remains constant, the coefficient increases as the depth diminishes. These experiments illustrate a curious point, namely, that various small orifices, 2 ins. thick (made in a 2-in. plank), run full like a short tube, .and these orifices therefore discharge more water than they theoretically should if the vein were contracted. The -in. X 2-in., i-in. X 2-in., and 2-in. X 2-in. orifices run full under these conditions, as also does the i-in. X i-in. orifice. The i-in. X 2-in. orifice, 2 ins. thick, is just on the margin between flow with contraction and full-bore flow. If it is fixed in the vertical position, with the longest diameter vertical, the vein contracts. If it is fixed in the horizontal position, with the longest diameter horizontal, it will also contract, but if it is rubbed with the fingers on the edge, it will run full for a time and then contract again. If kept running full in this way, it will discharge about I cu. ft. of water per minute more than when full contraction takes place. The 2-in. X 2-in. orifice runs partly full, that is to say, the lower half of the orifice, where the issuing vein curves down, runs full, while the upper half contracts. This largely in- creases both the discharge and the coefficient of discharge, but the flow becomes irregular and it is therefore practically impossible to measure a simple miner's inch. For this reason y 1 ^ of the flow from the 6-in. X 2-in. orifice was chosen as the standard for the unit miner's inch, and this miner's inch actually discharges 1.4982 cu. ft. per minute. 4 8 INVERSION OF THE JET. 14. Inversion of the Jet. The phenomenon of the inver- sion of the jet was first noticed by Bidone, and has been subsequently investigated by Poncelet, Lesbros, Magnus, Lord Rayleigh, the author, and others. Sectional Elevation. Cross-section. FIG. 31. FIG. 32. When a jet issues from an orifice in a vertical surface, the sections of the jet at points along its path assume singular forms dependent upon the nature of the orifice. With a square or rectangular orifice the section of the jet is a star of four sheets at right angles to the sides, Figs. 31, 32, 33- With a triangular orifice the section is a star of three sheets at right angles to the sides, Fig. 34- In general, with a polygonal orifice of n sides, the section is a star of n sheets at right angles to the sides. FIG. 33- FIG. 34. INVERSION OF THE JET. 49 These jets from non-circular orifices have central cores, and sheets at the edges are thickened out into beads, Figs. 33, and 34, which are approximately elliptical in section with major diameters double the minor diameters. Many exact measurements of these jets have been made and are partially described in a paper by Farmer and Strickland in the Trans. Can. Roy. Soc,, vol. IV. sec. 3. With a semicircular orifice the section has a more or less semicircular boundary and a single sheet at right angles to the diameter. The common explanation of this phenomenon is that the fluid particles issuing along different parabolic stream-lines impinge upon each other, and by their mutual reactions cause the jet to spread out and assume sectional forms depending upon the shape of the orifice. Thus the fluid particles issuing horizontally and freely at J3 f with a velocity \ f 2gAB, describe a parabola BD. The particle issuing at C with a "velocity describe a parabola CD of less curvature than BD. The particles cannot pass simultane- ously through the point D and must necessarily press upon each other. They are therefore com- pelled to move out of their natural paths, and the jet spreads into sheets. A theory which seems more FlG - 35- fully to account for the whole of the facts is that the peculiar changes in form are really due to surface tension and to the differences between the atmospheric pressure and the internal pressure of the jet. In the case, for example, of a jet flowing through an elliptical orifice with the major axis vertical, the stream-lines. 50 TIME OF FILLING A LOCK. in the vein are convergent and mutually react upon each other, causing the jet to contract vertically and elongate horizontally at a rate gradually increasing to a maximum, when the section is a circle in form. At this stage 'the rates of elongation and contraction are the same. The elongation and contraction still continue, but at a diminishing rate, until the movement is stopped by the effect of surface tension, when the section is again elliptical, with the major axis horizontal and the minor axis vertical. The new major and minor axes then again begin respectively to contract and to elongate, the section of the jet passing through the circular form to its initial elliptical form. This process is repeated over the whole length of the unbroken jet, and, in fact, in this portion of the jet the surface tension produces an effect similar to that which would be pro- duced if the jet were surrounded by an elastic envelope. If the orifice is small and the head not large, the jet, on leaving the- contracted section at the orifice, spreads out into sheets and then diminishes to a contracted section similar to the first, after which it again spreads out into sheets, bisecting the angles between the first set of sheets, and again diminishes to a contracted section. This action is repeated so long as the jet remains unbroken. A comparatively few experiments made in the laboratory indicate that if the head h is not large, the wave-length oc vV/ oc 7-. 15. Emptying and Filling a Canal Lock. When the liead varies, as in filling or emptying a reservoir or a lock, in filling a vessel by means of an orifice -under water, or in emptying water out of a vessel through a spout, Torricelli's theorem is still employed. If the lock or vessel is to be filled, Fig. 36, let X sq. ft. be the area of the water-surface when it is x ft. below the sur- face of the outside water. EXAMPLES. If the lock or vessel is to be emptied, Fig. 37, then JTsq. ft. is the area of the water-surface when it is x ft. above the orifice. In each case x ft. is the effective head over the orifice, and is the head under which the flow takes place. FIG. 36. FIG. 37. In the time dt the water-surface in the lock or vessel will ise or fall by an amount dx*. Then X . dx quantity which has entered the lock = cA \'2g* . dt, Xdx A being the area of the orifice. Hence t = CA 1/2gX an equation giving the time of filling or emptying the lock between the level x and h. The value of c for submerged orifices seems to be somewhat less than when the flow occurs freely, but it is usual to take .6 or .625 as a mean value. Ex. i. A paraboloidal vessel with a latus-rectum of i ft. and 5 ft. in height, is immersed in water to a depth of 4 feet. How long will it take to fill the Vessel to the level of the outside surface through an orifice i inch in diar. at the vertex ? (Take c = f .) Let 2y ft. be the diar. of the free surface when it is x ft. above the orifice. Then 52 EXAMPLES. Also, if the water rises dx ft. in dt sees., Tty*dx = amount entering vessel in dt sees. = quantity flowing through orifice under a head of 4 x ft. in dt sees. 57r -(4 ' 576 and therefore y . dx = xdx = -^-7(4 x)dt, or 576 .r 576 j 4 - (4 x) \ dt = -- -ax = - \ - - \ dx 5 (4 - *)* 5 I (4 - *)* \ Integrating between the limits ,r = o and x = 4 ft., the required time in sees. = 1228.8. Ex. 2. The horizontal section of a lock-chamber is approximately a rectangle and its length is 360 feet. The side walls have a batter of i in 12, and the width of the free surface when the lock is full of water is, 45 feet. How long will it take to empty the lock through two sluices in the gates, each 8 ft. by 2 ft., the sluice horizontal centre-line being 13 ft. below the free surface in the lock and 4 ft. below that of the canal on the down-stream side ? Let the level of the water in the lock fall x ft. in / seconds. The area of the water-surface is then If the level now sinks dx ft. in dt sees., 360(45 }d x = arnount of water which has flowed out through thet ' sluices = 2 . | . 2 . 8 . f/2 . 32 . X . dt = \6ox* ,2 2 - *WX o ' since, by the condition of continuity, Dividing throughout by the factor wa<, the equation becomes it W W g g \vhich may be written in the form +! = ^.4. A+i + feHUtf ~ l " W~T~ 2g ~ W^~ 2g^ 2g Now the pipes are nearly always axial, and in such case h = o, so that the last equation becomes ^ _i_ _ 1h 1 2g If there had been no abrupt change of section, or if the change between A'B 1 ' and CD had been gradual, then no internal work would have been done in destroying the parallel- ism of the fluid filaments, and no energy wasted. Therefore, by Bernoulli's theorem, the relation , ,., + + " l "*" / T 2^- ~ ' ' C ' 2 W 2g would have then held good. ( v V ) 2 Thus -- 2 ft.-lbs. of energy per pound of fluid is the Joss in shock between A'B' and CD. Experiment justifies the assumption P = p r BORDA'S MOUTHPIECE. Ex. At a point A, 150 ft. above datum, a line of piping suddenly doubles in sectional area. If the velocity of flow in the larger pipe is 8 ft. per sec., and if the pressure *it A is 125 Ibs. per sq. in., find the pres- sure per sq. in. at B, 8 ft. above datum, the motion being steady. The velocity of flow in the smaller pipe is evidently 16 ft. per second Therefore the loss of head in shock at the sudden change of section (16 - 8)* 2.32 = i ft. Hence, if/ is the pressure per sq. in. at B, p x 144 8* 8 + '-*-? + ,64 + ' = 15 x 144 i6 or and , 144 *' 62* / = 432, - in - 18. Mouthpieces. (a) Borda s Mouthpiece. This is merely a short pipe projecting inwards, as in Fig. 39, which FIG. 39- represents a jet flowing through a re-entrant mouthpiece of sec- tional area A, fixed in the vertical side of a vessel of constant horizontal section and containing water kept at a constant level. The mouthpiece is sufficiently long to allow of the jet springing clear from the end EF without adhering to the inside surface. BORDJ'S MOUTHPIECE. 59 The velocity of the fluid molecules along AC and DK, is sufficiently small to be disregarded, so that the pressure over this portion of the vessel is distributed in accordance with the hydrostatic law. The same may also be said of the pressure on the remainder of the vessel's surface. Again, the only unbalanced pressure is that on the surface HG immediately opposite the mouthpiece, and the resultant horizontal force in the direction of the axis of the mouthpiece = (A + WJ i) A ~ PV& = kA > k being the depth of the axis below the water-surface and / the intensity of the atmospheric pressure. The jet converges to a minimum, or contracted section MN, of area a. In a unit of time let the fluid mass between AB and MN* take up the position bounded by A 'B and M'N'. Then wkA = impulse of force in direction of motion = change of momentum in same direction in a unit of time = difference between the momenta of MNN'M' and ABB'A\ since the momentum of the mass between A'B' and MN remains unchanged = momentum of MNN'M', since the momentum of ABBA' is vertical w = Z) - / . area J^F + w . area 677. GK . ' , very nearly = whA. Hence, in a unit of time, whA = impulse of this force = change of momentum in direction of axis w w w = av . v = air = -- a . 2gh, a being the area of the contracted section, while // is also very approximately the depth of its C. G. below the water-surface. Thus, as before, the coefficient of contraction = ;- = . A 2 (&) Ring-nozzle. The ring-nozzle (see Fig. 41) is often used with a fire-engine jet, and consists of a re-entrant pipe of sectional area a l fixed in a pipe of sectional area a r The length of the re-entrant portion is such that the water springs clear from the inner end and, without again touching the surface of the mouthpiece, FlG> 41 - converges to a minimum or contracted section of area a at MN. Consider the fluid mass between MN and a transverse sec- tion AB, and in a unit of time let it move into the position bounded by the planes ~M'N' and A ' B' . 62 RING-NOZZLE. It is assumed that the motion is steady and that there is no internal work due to the production of eddies or other motions. Let/> , v be the intensity of the atmospheric pressure and the velocity at MN. Let p l , 7>j be the mean intensity of pressure and the velocity at AB. Let P be the mean intensity of the pressure over the annular surface EF, GH. Let ,8*0, s l be the elevations above datum of the C.-G.s of the sections MN and AB. Then impulse -in direction of motion change of momentum in same direction in a unit of time = difference of the momenta of the fluid masses MNN'M' and ABB' A' h + * ' * W 2g and w 2g a v where c = coefficient of contraction = = . Therefore: A ' it, Po - *>= , T w .; (4) an equation giving the velocity of flow at the point of discharge^ CYLINDRICAL MOUTHPIECE. 65 If the discharge is into the atmosphere, p = p and equa- tion (4) becomes V 2 = - - -- rj = C.' . 2gh, ... ( 5 ) i where If C,, = .62, then c v = .85, while experiment gives .82 as the value of C v . The small difference between .85 and .82 is probably due to frictional resistance. The value ..82 for c v makes C. approximately .617. Again, the discharge from a simple sharp-edged orifice of same sectional area as the mouthpiece is .62 A \/2gh, or more than 24 per cent less than the discharge from the cylindrical mouthpiece. The loss of head between MNund M'N' ? (r *) &" \r ^ , and therefore (2) or, introducing a coefficient c v (= .98, nearly, for a smooth well-formed mouthpiece), , ...... (3) and the discharge is = c v A (4) DIVERGENT MOUTHPIECE. 67 From the last equation it would appear as if the discharge Avt>uld increase indefinitely with A, but this is manifestly im- possible. In fact, by eq. (i), the flow being into the air, and taking W W since av^ = Av. But/ L cannot be negative, and therefore so that A /1T~ wh + i ...... (7) A / a V gives a maximum limit for the ratio of A to a. Now -- = 34 ft. very nearly, and the last equation may be \V^ written By eqs. (4) and (7), 0=c,ay/2g(h + ^) ) ... (9) which is also the expression for the discharge through the minimum section a into a vacuum. If, however, the sectional areas of the mouthpiece at the point of discharge and at the throat are in the ratio of A to a, as given by eq. (7), it is found that the full-bore flow will be interrupted either by the disengagement of air, or by any slight disturbance, as, for example, a slight blow on the 68 CONVERGENT MOUTHPIECE. mouthpiece, and hence, in practice, it is usual to make the ratio of A to a sensibly less than that given by eq. (7). (e) Convergent Mouthpiece. With a convergent mouthpiece (Fig. 44) two points are to be noted : (i) There is a contraction within the mouthpiece, followed by a swelling out of the jet until it again fills the mouthpiece. FIG. 44. Thus, as in the case of cylindrical mouthpieces, there is a ' ' loss of head ' ' between the contracted section and the point of discharge, and also a consequent diminution in the velocity of discharge. (2) There is a second contraction outside the mouthpiece due to the convergence of the fluid filaments. The mean velocity of flow (v 1 ) across the section is Cy being the coefficient of velocity and h the effective head above the centre of the section. Also, the area of this section == c c ' X area of mouthpiece at point of discharge ct.A, c c f being the coefficient of contraction. Hence the discharge Q is given by Q = c v 'c c 'A V^rk = C 'A '(= c v 'c c ') being the coefficient of discharge. ENERGY AND MOMENTUM OF JET. 69 The coefficients c v f and c depend upon the angle of con- vergence, and Castel found that a convergence of 13 24' gave a maximum discharge through a mouthpiece 2-6 diameters in length, the smallest diameter being .05085 foot. TABLE GIVING CASTEL'S RESULTS. Angles of Convergence. C ' c C ' V C' Angles of Convergence. C c be the vel. at the mouth. Then /_ + *'_= 34 + ^ = 36. W 2g ^ ^ 2g * V * Therefore ~ 2, or v = 8 |/2 ft. per sec. 72 RADIATING CURRENT. The velocity in the throat is greatest when the pressure, pi, is least, i.e., when p* = o, and then v* o H -- = 36, or v 48 ft. per sec. A If D ins. is the diameter of the mouth, the discharge 144 4 144 4 n = cu. ft. per sec., H and D* = 12.726, or D = 3.56 ins. 20. Radiating Current. As an application of Bernouilli's theorem, consider the steady plane motion of a body of water flowing radially between two horizontal planes a ft. apart, and symmetrical with respect to a central axis (Fig. 46). Let v ft. per second be the velocity at the surface of a cylinder of radius r ft. described about the same axis. Then the volume Q crossing the surface per second is Q = 2nr . av y and therefore rv = - = a constant, since Q is constant. Thus v increases as r diminishes, and becomes infinitely great at the axis ; but it is evident that the current must take a new course at some finite distance from the axis. If p is the pressure at any point of the cylindrical surface z ft. above datum, then, by Bernouilli's theorem, p V* v* z -I -- 4- = a constant //=.- y 4- r w ^ 2 i RADIATING CURRENT. 73 denoting the dynamic head z -j by y. Hence a constant 2 so that vr = a constant, and v varies inversely as r, as in the case of the radiating current. Therefore the curves of equal pressure will also be the same as in a radiating current. Free Spiral Vortex. Suppose that the motion of a mass of water with respect to an axis O is of such a character that at any point M, the components of the velocity in the direction of OM, and perpendicular to OM, are each inversely propor- tional to the distance OM from O. The motion is thus equiva- lent to the superposition of the motions in a radiating current and in a free circular vortex ; and if is the angle between OM and the direction of the stream-line at M, v cos 8 and v sin 6 are each inversely proportional to OM, and therefore # must be constant. Hence the stream-lines must be equi- angular spirals, and the motion is termed a free spiral vortex. This result is of value in the discussion of certain turbines and centrifugal pumps. A steady free surface in the case of a free spiral vortex is impossible, as the stream-lines cross the surfaces of equal presure, which are the same as before. Also, if p Q , r , 7' are the pressure, radius, and velocity at any other point at the same elevation z above datum, then - , i , fi : !, + A + !i* ' W ' 2g W 2g" and the increase of pressure-head Forced Vortex. A forced vortex is one in which the law of motion is different from that in a free vortex. The simplest and most useful case is that in which all the particles have ar\ equal angular velocity, so that the water will revolve bodily, 76 FORCED AND COMPOUND VORTICES. the velocity at any point being directly proportional to the distance from the axis. As before, d L , A = ^ \ '*~ w ) - g r ' But v oc r = oor, co being the constant angular velocity of the rotating mass. Therefore Integrating, d\z + - J = r . dr. \ I nntl fr v* , 2 -] -- - - +'a constant = --- 1- a constant. r W 2g 2g Hence, if / , r , ^ are the pressure, radius, and velocity for any second point at the same elevation z above datum, then W If the second point is on the axis of revolution, then r = o, and th.e last equation becomes P-P, \ W 2g Thus the free surface of the pressure columns is evidently a paraboloid of revolution with its vertex at O, as in Fig. 48. A compound vortex is produced by the combination of a central forced vortex with a free circular vortex, the free sur- face being formed by the revolution of a Barlow curve and a parabola. For example, the fan of a centrifugal pump draws the water into a forced FIG. 48. vortex and delivers it as a free spiral vortex into a whirlpool - chamber (Chap. VIII). EXAMPLE. 77 In this chamber there is thus a gain of pressure-head, and the water is therefore enabled to rise to a corresponding addi- tional height. James Thomson adopted the theory of the compound vortex as the principle of the action of his voitex turbine. Ex. A centrifugal pump of 2 ft. interior and 4 ft. exterior diar., ma^es 336 revols. per minute. The water gradually fills up and flows very slowly through the wheel into a chamber of comparatively much larger diar., from which it passes away into the discharge-pipe. The pressure at the inlet may be taken to be one atmosphere, or 2116 Ibs. per sq. foot. Basing the flow through the wheel upon the hypothesis that the velocity v of any fluid particle is directly proportional to its distance r 84.35 FIG. 49. from the axis of rotation, the law connecting the pressure p and the velocity v may be expressed in the form (Ex. I, p. 21) W 1g At the inlet/ = 2116, and let v v\. . Then 2116 v? so that But p 2116 125/336 22 V T? r* = ^ 2 ) = 1210, and - = 128\ 60 7 j Vi* I 2 7 8 LARGE ORIFICES. Therefore p = 2116 + 1210 (>' i) = 906 + i2ior a , Giving r, successively, the values i, 1.2, 1.4, 1.6, 1.8, 2ft., the corresponding values of /are 2116, 2648.4, 3277.6, 4003.6, 4826.4, 5746165. Thus the curve AB, obtained by plotting these values, shows the variation of the pressure inside the wheel. The hypothesis of the flow in the surrounding chamber is that the velocity of any fluid particle is inversely proportional to its distance from the axis of rotation ; and in this case the pressure and velocity are con- nected by the relation (Ex. i, p. 21) p v* W 2g' At the wheel outlet, i.e., where r = 2 ft.,/ = 5746 Ibs. per sq. ft., and let v = v*. Then 1746 = c _^. W 2g* therefore p = 5746 + ^(i - --} 2g \ V**} But therefore 125 /3^6 22 \' 7/2 4 = 4840, and - = -> 1 28 \ 60 7 7 T/ 2 r ' p = 5746 + 4840(1 - i} = 10586 - -^ Giving r, successively, the values 2, 2.2, 2.4, 2.6, 2.8, and 3ft., the corresponding values of /are 5746, 6586, 7225, 7723, 8117, and 8435 Ibs. Thus the curve BC, obtained by plotting these values, shows the va- riation of the pressure in the chamber surrounding the wheel. 22. Large Orifices in Vertical Plane Surfaces. The issuing jet is approximately of the same sectional form as the orifice, and the fluid filaments converge to a minimum section as in the case of simple sharp-edged orifices. (a) Rectangular Orifice (Fig. 50). Let E, F be the upper and lower edges of a large rectangular orifice of breadth LARGE RECTANGULAR ORIFICES. 79 J?, and let H l , // 2 be the depths of E and F, respectively, below the free surface at A If u be the velocity with which u* the water reaches the orifice, then H , is the fall of free ig surface which must have been expended in producing the velocity u. Hence H v -\- H and 7/ 2 + H are the true depths of the edges E and F below the surface of still water. Let MN be the minimum or contracted section, and assume that it it is a rectangle of breadth b, Let h l , // 2 be the depths of M and N, respectively, below the free surface at A. Then /^ + //, h z + H are the true depths of M and N below the surface of still water. First. Let the flow be into the air, the orifice being clear above the tail-water level, Fig. 50. Consider a lamina of the fluid at the section MN, of the 4, lr _.;_.,__ .t_. .i_ i, FIG. 50. width of the section, and between the depths x and x + dx below the surface of still water. The elementary discharge dq, in this lamina, is dq = bdx \/2gx, 8o LARGE RECTANGULAR ORIFICES. and therefore the total discharge Q across the section MN is Q /fkt + dq = I b . J hi + dx + H e = Then IT)*}.. The coefficient ^ is by no means constant, but is found to vary both with the head of water and also with the dimensions of the orifice, and can only be determined by experiment. Second. Let the orifice be partially (Fig. 51) submerged, and let H " 3 be the depth between the surface of the tail-race water and the free surface at A. By what precedes, the discharge Q l through EG, the portion of the orifice clear above the tail-race, is -(/f , + #)*}. (2) Every fluid filament flows through the portion GF of the orifice under an effective head H z + H, and there- fore with a velocity equal to FIG. 51. Hence the discharge Q t through GF is ', ... (3) and the total discharge Q is equal to Q l -\- Q 2 . LARGE RECTANGULAR ORIFICES. 81 -, The coefficients c l , c 2 are to be determined by experiment, and if c l c 2 = c, (4) Third. Let the orifice be wholly submerged (Fig. 52). Then the total discharge Q is evidently H, (5) c being a coefficient to be determined by experiment. If the velocity of approach, u, is sufficiently small to be disregarded without sensible error, then H o, and equations (i), (4), and (5), respec- tively, become Q= -cB J - H?}; (6) (7) (8) (b) Circular Orifices. Let Fig. 53 represent the minimum section of the circular jet issuing from a circular orifice. Let 26 be the angle subtended at the centre by the fluid lamina between the depths x and x + dx below the surface of still water. Let r be the radius of the section so that 2r = h 2 h^ , h^ and h^ being, as in (a), the depths of the highest and lowest points of the orifice below the free surface at A. 82 LARGE CIRCULAR ORIFICES. H, as before, is the head corresponding to the velocity of approach. FIG. 53- Then the area of the lamina under consideration = 2r sin 8 . dx, and the elementary discharge, dq, in this lamina, is dq 2r sin . dx h^ + H+h^ + H /i, But x = - - J - J -- r cos = - 1 r cos 0, and therefore Hence dx = = 2 r sn 277 r cos and the total discharge (2 is r cos (9) Ex. The free surface on the up-stream side is 5 ft. and on the down- stream side i ft. above the sill of a rectangular sluice 12 ft. wide. How much must the sluice be raised to give 105,000 gals, per minute ? 105,000 105,000 gals. per. mm. = = 280 cu. ft. per sec. NOTCHES AND WEIRS. 83 Let x ft. be the opening above the sill. For a depth of i ft. above tne sill the discharge is under a constant head of 5 i = 4 ft. For the remainder of the opening the discharge takes place freely through a rectangular orifice, with its upper and lower boundaries respectively <5 x} ft. and 4 ft. below the up-stream surface. Then 280=. 12. 1. . 4* +7. 3 -(5 - x}\ + 4!} = 440 40(5 x)*. Therefore (5 *)* = 4, and x = 2.48 ft. 23. Notches and Weirs. When an orifice extends up to the free-surface level it becomes what is called a notch. STILL WATER LEVEL FIG. 54. FIG. 55. A weir is a structure over which the water flows, the dis- charge being" in the same conditions as for a notch, and is very useful for gauging the flow of small streams, the amount of water supplied to hydraulic motors, etc. Rectangular Notch or Weir. The discharge may be found by putting H 1 = o. Thus equation (i) becomes Q = jc^i(ff s +.ff)*-ff*}. . . (10) If the velocity of approach be disregarded, then H ~ o, and the last equation becomes (ii) 8 4 WEIRS. and H 2 is the depth to the bottom of the notch or to the crest of the weir. Great care should be taken in obtaining the accurate value of H z . A hook or a stiff vertical rod, with a sharp point, may be fixed, at a suitable distance (5 to 8 ft.) from the back of the weir, with the point on a level with the crest of the weir. The flume is then filled with water rising slightly above the crest and producing a capillary elevation of the surface at the point. The water is now allowed to subside until the eleva- tion is barely perceptible, when a hook-gauge (Chap. Ill) is adjusted and a reading taken. A second reading is taken for any required discharge over the weir, and the difference between the two readings is the depth, // 2 , of the water on the crest. It has been found that the discharge (Q) is appreciably affected by vibration, and it is therefore of importance that the weir should be made as rigid as possible. The up-stream face of the weir is nearly always vertical and at right angles to the direction of flow. To diminish the effect of the velocity of approach, the water-section in the flume should be large as compared with the section of the waterway on the crest, and the depth of the weir should therefore be at least twice the depth H 2 of the water on the crest. The crest should be horizontal and, generally speaking, it consists of a plate with a bevelled edge, Fig. 54, on the up-stream side, or of a thin plate, 'Fig. 55, so that the water springs clear from the inner edge. A rounded edge, Fig. 56, diminishes the discharge and should be avoided, as its effect is uncertain. WEIRS. The length B of the crest should be at least three times the depth H Y The effective sectional area of the water flowing through a rectangular notch, or over a weir, is less than BH 2 , because of (a) crest contraction, (b) end contraction, (c) the fall of the free surface towards the point of discharge. It is reasonable to assume that the diminution of the actual sectional area, BH 2 , due to crest contraction and to the fall of the free-surface level is proportional to the width B of the opening. Suppressed Weir, or Weir without End Contractions. If .a weir occupies the whole width of the stream, or flume, Figs. 57 and 59, the contraction at each end is wholly suppressed, FIG. 57. FIG. 58. FIG. 59. and crest contraction only takes place, i.e., the falling sheet of water is reduced in thickness near the crest. Air must be freely admitted below the falling sheet, as otherwise a partial or complete vacuum will be produced and the sheet will be depressed or will adhere to the face of the weir, while the dis- charge Q will be very sensibly modified. Francis effected the free admission of air and also prevented the lateral spreading of the sheet, after leaving the crest, by prolonging the upper portions of the flume sides a short distance beyond the weir, Fig. 58. The discharge was thereby diminished by about .4 per cent. 86 WEIRS. Weir with End Contractions. These contractions occur when the sides of the weir, or notch, Figs. 60 and 61, are at a distance from the sides of the channel, and they have the FIG. 60. FIG. 61. effect of diminishing the discharge. The contraction is com- plete, i.e., as great as it can be made, when the distance of the weir side from the channel side is not less than about the depth H 2 . Other things being equal, the contraction and its effect upon the discharge increase with H 2 . The effect of end con- tractions is almost inappreciable and may be disregarded when the length B of the crest is not less than about H z ; but as the r> ratio -fj- diminishes, the effect rapidly increases. Francis ^2 found that the discharge for a weir with perfect end contrac- tions and in which B = 4// 2 , was diminished 6 per cent. In his Lowell weir experiments he also found that, for depths //" 2 -[- //"over the crest varying from 3 ins. to 24 ins., and for widths B not less than three times the depth, a per- fect end contraction had the effect of diminishing the width of the fluid section by an amount approximately equal to one- tenth of the depth, or B - 10 so that the effective width = 10 Thus, if there are n end contractions, the effective width B (H 2 -f- H), and the equation giving the discharge WEIRS. 87 becomes . (12) According to Francis the average value of c in this equa- tion is .622. Then \c ^2g = 3.33, very nearly, and therefore Q = 3-335 - (//; + H)(H t + //)* - /tf. . (13) In experiments carried out by Fteley and Stearns with suppressed weirs, as described above, the total variation in the value of the coefficient was found to be about '2-J- per cent. The depths H 2 were measured 6 ft. from the weir, and for values of H 2 exceeding .07 ft. they deduced the formula (2 = ^(3.31^2* +.007), in which the velocity of approach is disregarded. Allowance may be made for the velocity of approach by substituting for H 2 the expression HI~\- \\H according to Fteley and Stearns, but H 2 -\- i^H according to Hamilton Smith, Jr., who bases his conclusions upon a comparison of the experiments of Fteley and Stearns with those of Francis and others. f-f I f-f If the weir has n end contractions, B n ^^- must be 10 substituted for B, and allowance is made for the velocity of approach by substituting for H 2 the expression //" 2 -f- 2.05/7, according to Fteley and Stearns, or H z -f- 1.4/7", according to Hamilton Smith, Jr. Hunking and Hart give the formula 88 SUBMERGED WEIRS. in which /* is very nearly = I -| (~^ 2 ) > where S = sectional area of waterway 10 Bazin gives the formula Q = c in which, if the velocity of approach is disregarded, . .00984 c = .405 + -rfA "i but if allowance is to be made for the velocity of approach, x being the height of the weir. Bazin considers that, with suppressed weirs, as already described and which are not very low, the results obtained with this coefficient are accurate within I per cent. Submerged, or Drowned, Dams (or Weirs), In these the surface of the tail-race water rises above the top of the dam, STILCWATER LEVEL FIG. 62. Fig. 62. It may be assumed that between a and b the flow is as over the crest of a weir, the depth of water on the crest INCLINED WEIRS. 9 being // 2 -j- //, and that between b and c the flow is equiva- lent to that through a submerged orifice under a constant head H^ _|_ H, Hence, if H' is the depth of the top of the dam below the surface of the tailwater, and if c is the coefficient of discharge both for the flow between a and b and also that between b and c, Q=-c o The following table gives approximate values of c corre- TTf sponding to different values of the ratio -y r i as deduced from experiments carried out by Francis, the head over the crest varying from I to 2.32 ft., and by Fteley and Stearns, the head varying from .325 to .815 ft. : Values of H 1 Corresponding 4 Values of c as deduced from the experiments of Fteley and Stearns. .625 to .635 Hi + H+H' 0C . v rar 623 t LC1S. o .632 ' .630 '.625 .615 1 .610 ' .607 ' .607 .607 1 .607 620 20 610 an t;a8 dO 586 en egc 60 , .e8q .70. . ... .$8$ 80 egc QO .05. . .618 .600 .590 .585 .583 .580 .581 .590 .610 .628 .610 .600 595 593 590 591 .600 ' -615 ( Trautwine.} Inclined Weirs. If the up-stream face of a weir, instead of being vertical, is inclined up-stream, Fig. 63, the discharge is diminished, the depression of the upper surface of the falling sheet of water commences near the crest, while the lower sur- face rises higher, above the crest, and moves backwards. If the face is inclined down-stream, Fig. 64, the discharge is increased, the depression of the upper surface commences at a point farther from the crest than when the face is vertical, CIRCULAR NOTCH. while the lower surface becomes more flattened and moves away from the weir. Values of the coefficient of discharge for inclined weirs have been deduced by Bazin and are given in a subsequent article. FIG. 63. FIG. 64. The discharge is increased by rounding the up-stream edge of the weir. Circular Notch. In equation (9), Art. 22, put h^ = o and h n = 2r. Then Q = 2r 2 sin 2 0ff+ 2r sin 2 -)V0, and if the velocity of approach be disregarded, so that H o, //i sin* 0\ sin -dO r (2 sin - - sin ^ + si * sn 64 -- 15 Ex. i. A dam with a rectangular notch 6 ft. wide is formed across a channel; and the depth of the water over the sill is 12 ins. Find the quantity of flow when the notch has (a) no side contraction ; (&) one side contraction ; (?) two side contractions. EXAMPLES. 91 Disregarding the velocity of approach, and assuming the coefficient of discharge to be the same in each case, viz., f, (a) Q 1 = . 1/64 . if . 6 = 20 cu. ft. per sec. 0) fit = y J 4/64. Ia '( 6 ~ A) = 1 91 cu. ft. per sec. (c) Q t = ~.l. 4/64 . ii(6 - A) = io cu. ft. per sec. Ex. 2. 400 cu. ft. of water per second are conveyed by a channel of rectangular section 25 ft. wide, when the water runs 4 ft. deep. Find the height of a dam built across the channel which will increase the depth 50 per cent, taking into account the velocity of approach. 400 8 The velocity of approach = - 7 = - ft. per sec. /8\2 j The corrresponding head = ~ = - ft. Let x ft. be the height of the dam. First. Assume that the dam is not drowned, i.e., that its crest rises above the water-surface on the down-stream side. Then 400 = y.. or (6 x + 1)1 = 4.837037 = (6.1 1 1 *), and x 3.25 ft., which is less than 4 ft., and therefore the assumption that the dam is not drowned is incorrect. Second. Assuming that the dam is drowned, the discharge now takes place under a constant head of (2 -f ) ft. for a depth of (4 x} ft., and as over a weir for a depth of 2 ft. Then 400 = | . 25(4 - *) 4/6^(2 + l)*+'y { 25 . 4/64. {(2 + 1)1 - ft)!}, or 1.1798 = (4 *)(2l)* and x = 3.188 ft., which is less than 4 ft., and therefore the assumption of a drowned dam is correct. Ex. 3. If x is the depth of water over the crest of a rectangular notch, then, disregarding the velocity of approach, Q = 9 2 TRIANGULAR NOTCH. Let dQ be the change in the discharge corresponding to a change dx in the depth on the sill. Then . -.r* . x + dx. The area of the lamina FIG. 65. and the discharge in this lamina is Hence the total discharge Q is Q = c = cB cu. ft. per sec. (14) c is a coefficient introduced to allow for contraction, etc., and Professor James Thomson gives .617 as its mean value for a sharp-edged triangular notch. EXAMPLES. 93 T) %Now the ratio -77- is constant in a triangular notch and 2 varies in a rectangular notch. Hence Thomson inferred and showed by experiment that the value of c is more uniform for triangular than for rectangular notches, and therefore also the former must give more accurate results. If the flow is through a 90 notch, B = 2// 2 , and 5 Q = c \ / 2gH^ = 2.64//" 2 * cu. ft. per sec., approximately, or = 158.385/^2^ cu. ft. per min., c being .617 and g = 32. 176. Ex. i. A reservoir discharges through a sharp-edge triangular notch, and in/ sees, the depth of the water in the notch Jails from H ft. to x ft. Let S be the sectional area of the reservoir corresponding to the x ft. depth ; let mx be the width of the free surface on the notch corre- sponding to the x ft. depth, m being a numerical coefficient depending up>on the notch angle. Then, since the water sinks dx ft. in dt sees., 5. dx = discharge from reservoir in dl sees. = amount flowing through notch in dt sees. = Vzgc - mx*. dt, or dt = --- x ~ . dx. 4 Hence the time in sees, in which the depth falls from H ft. to x ft. 4 If the horizontal sectional area 6" is constant, 55 / I i \ the time in sees. = - 5 -- -77^ I. For a 90 notch m = 2, and taking^ = 32 and c = f, SI \ i \ the time in sec*. = --p _ _J. 94 BROAD-CRESTED WEIRS. The time becomes infinite when x = o, which indicates that the flow diminishes indefinitely with the depth in the notch. Ex. 2. Find the discharge in gallons per minute through a 90 sharp- edge notch when the water runs 4 ft. deep. If the reservoir supplying the water has a constant horizontal sectionalarea of 80,000 sq. ft., in what t-ime will the level sink 3 ft. ? Q = ~ ^64 . |- . 2 . 4 s = 85^ cu. ft. per sec. = 85* x 6* x 60 gals, per min. = 32,000 gals, per min. the time ( i > ) = 1 7,S secs - = 4** hours. 4V 4* / 25. Broad-crested Weir. Let Fig. 66 represent a stream flowing over a broad-crested weir. On the up-stream side the FIG, 66. free surface falls from A to B. For a distance BD on the crest the fluid filaments are sensibly rectilinear and parallel; the inner edge of the crest is rounded so as to prevent crest con- traction. Consider a filament ab, the point a being taken in a part of the stream where the velocity of flow is so small that it may be disregarded without sensible error. Let A be the thickness MN of the stream at b. Let the horizontal plane through N be the datum plane. Let z^ , .s- be the depths below the free surface of a and b. Let h be the elevation of a above datum. BRO4D-CRESTED WEIRS. 95 -, Let p Q , p^ , / be the atmospheric pressure and the pressures at a and b. Let v be the velocity of flow at b. Then, by Bernouilli's theorem, But 4 = ^+4 and ^ = *+4; W ' 2f W ' W therefore IV 10) 2ff and hence V* 2f i + *i- =X 2 -^ H 2 being the depth of the crest of the weir below the surface of still water. Thus, if B be the width of the weir, the discharge Q is - A). .... (16) From this equation it appears that Q is nil both when A = o and when A = H v Hence there must be some value of A between o and H 2 for which Q is a maximum. This value may be found by putting d Q = = B ^ g ( VH^ - ~^^, and therefore 9 6 BROAD-CRESTED WEIRS. and the expression for the discharge becomes Q = BH * S&** = -385^ 2&ff, (17) which is the maxmum discharge for the given conditions. Experiment shows that the more correct value for the dis- charge is . ..... (18) If the water approaches the weir with an appreciable velocity U* u, corresponding to the head H, so that - //, then and This formula agrees with the ordinary expression for the discharge over a weir as given by equation (n), if = -525. It might be inferred that for broad-crested weirs and large masonry sluice-openings the discharge should be determined by means of equation (18) rather than by the ordinary weir formula, viz., equation (n). It must be remembered, however, that in deducing equa- tion (17), frictional resistances have been disregarded and the gratuitous assumption has been made that the stream adjusts itself to a thickness / which will give a maximum discharge. The theory is therefore incomplete. The discharge over a sharp-crested weir is sensibly the same as that over a weir with an apron, as in Fig. 66, so long as the depth of the water on the crest is not less than about 15 ins., but below this limit, the discharge over the apron RESERVOIR SLUICES. 97 rapidly diminishes with the depth. For example, the dis- charge over a sharp-crested weir is approximately double that over a weir with an apron when the depth is about I in., is 20 per cent greater when the depth is 6 ins., and 10 per cent greater when the depth is 12 ins. 26. Reservoir Sluices. The water flows into the receiving ^channel either freely, as in Fig. 67, or under water, as in Fig. 68. FIG. 67. In the first case, the stream-lines converge to a contracted FIG. 68. section, and between the sluice and a certain section DE there is a sudden swell, the height of swell being given by 7, 2 7, 2 EC -i- - -* J-) \s ' and v l , ^ 2 being the velocities of flow across the contracted section and the section at DE. Let A lt A 2 be the areas of the sluice and section at and let nA = A . Then 98 RESERVOIR SLUICES. c c being the coefficient of contraction. Thus the swell will be found to be further from or nearer the sluice, according as the difference between the depths of the stream and the sluice is > or < BC. If/ 2 is the coefficient of hydraulic resistance, then and/ 2 may be . I or even greater; but if the sluice edges are smoothed and rounded so that/ 2 can be disregarded, then and therefore AB - BC = AC = ^-. *g It is assumed that the water in the reservoir retains the same level, but where the flow commences there is a depres- sion in the surface due to the velocity of flow, and the amount of this depression should be deducted from the total head. When the backwater rises above the sluice, as in Fig. 68, AC = head required to produce v 2 -\- head " lost in shock " f 2 ^2 and AC increases with , i.e., as A l diminishes as compared with A n . FLOW OVER WEIRS (BAZIN}. 99 27. Bazin's Flow Over Weirs. This article is the resume of Bazin's val- uable papers on this subject published in the Annales des Fonts et Chaussees. The symbols are changed to correspond with the preceding; articles of the present chapter. Let c s , s , H s be the coefficient, length of crest, and head over crest for a standard weir. Let c, B, H* be the corresponding symbols for an experimental weir, Then, disregarding the velocity of approach, and Experiments with the standard weir give the value of c s , the ratio n J-J s is usually unity, and the ratio - is found by observation. Hence Jt> J~ii the value of c can be at once calculated. In practice it seems impossible, with the data at present available, to make a rational selection of the proper value of c, which varies betweert wide limits and is affected not only by the form of the weir but by other conditions, amongst which may be enumerated the following : (a) The velocity of approach, which cannot be disregarded when the weir is of small height. (ft) The height of the weir. ( o^.io, it is sufficiently accurate to take .oo-i H = .405 + -, FLOW OYER WEIRS (BAZ1N}. 101 and therefore so that Generally, for values of // 2 between o m .io and o m .3o, jj. may be made- equal to .425, and, taking K = .5, a suitable form for practical use when errors of 2 to 3 per cent are not too large to be of importance. The absolute values of c having been found for a sharp-crested weir with a free nappe and a vertical face on the up-stream side, it does not follow that the same method should be adopted to determine the cor- responding coefficients for other forms of weir. In fact, if c' is the coef- ficient for any other given weir, when the head over the crest is the same, the influence of the velocity of approach may be largely eliminated by finding the ratio . The ratio corresponding to two different in- clinations is sensibly constant for all heads, and the following table gives the values of - for varying face-slopes: For an up-stream face-slope of i hor. to I vert ....... = .93 2 " 3 " ...... " = .94 i " 3 " ....... " = -96 " " vertical face ...................... " = .00 For a down-stream face-slope of I hor. to 3 vert ..... " = .04 2 " 3 "...."= .07 1 I " ____ " = .10 2 I " ____ " = .12 4 " I "...."= .09 It may be noted that the coefficient (or ratio) gradually increases from .93, corresponding to a slope of 45 on the up-stream side, to 1.12, corresponding to a slope of about 30 on the down-stream side. 302 FLOW OVER WEIRS (BAZIN). When the air cannot pass uuderneath the sheet of water flowing over the crest, the nappe either encloses a volume of air at less than the atmospheric pressure and is depressed, Fig. 69, or the air is entirely ex- FIG. 69. Depressed Nappe. FIG. 70. Drowned Nappe. eluded and the nappe is wetted underneath or drowned, Fig. 70. The latter condition, when the nappe encloses an eddying mass of fluid, gives a more uniform motion, as the pressure of an enclosed volume of air may vary from the accidental admission of new air. The discharge is slightly greater than with the free nappe, and may be increased almost 10 per cent when the nappe is on the point of being drowned. So long .as the head exceeds a certain limit, the nappe will not be in contact with the weir face. The drowned nappe may be either independent of or influenced by the down-stream level according as the rise produced beyond the. nappe is at a distance from the foot of the nappe or partially -encloses the foot. Rise at a Distance from the Foot of the Nappe. In this case - = .878 + .128 , c ^ Hi t>ut the max. value of cannot exceed 2|, as the drowned condition no //a longer holds when // 3 < " x. The value of --. corresponding to this maxi- mum, is 1.2, and if /7 a = x, the coefficients c' and rare sensibly the same. Applying this formula to weirs of different heights, it is found that the absolute values of the coefficients of discharge are sensibly given by the formula * c' = .47 Rise Enclosing the Foot of the Nappe. if D is the difference of level ^between the weir-crest and the down-stream surface, FLOW OfER WEIRS (BAZIN). 103 for which it is usually sufficiently accurate to substitute the simpler expression c' D These formulae are only true for values of D between certain limits. If Hi + D is greater than about \x, the rise is moved beyond the foot of the nappe, and the formulae in the preceding case become applicable. Again, if the head Hi is not sufficient to enable the nappe to push back the rise, the down-stream surface level must be sufficiently high to pre- vent the admission of air below the nappe. The drowned nappe preserves its characteristic profile even when the down-stream surface is on a level with the weir crest, Fig. 71, but if the difference of level between the up- and down-stream surfaces still continues to diminish, a point is reached at which the nappe suddenly and with an undulating movement again forms part of the surface. This change, which is very apparent, does not seem to have much influence on the coefficient of discharge. FIG. 71. Drowned Nappe. FIG. 72. Adhering Nappe (ff t not very small). FIG. 73. Adhering Nappe, spring- ing clear above crest. FIG. 74. Adhering Nappe small). On certain rare occasions, and under conditions governed by the thickness of the weir and by the construction of the upper portion carry- ing the crest, the nappe becomes adherent, Figs. 72 to 76. the sheet of water remaining in contact with the weir face. The coefficient c' is then 104 FLOW OVER WEIRS (BAZ1N). increased and may become as large as 1.3^, corresponding to an absolute- value of .55 or .56. From what has been said it may be at once inferred that the dis- charge over a weir is largely influenced by the form of the nappe_ FIG. 75. Nappe adhering on crest only. FIG. 76. Taking, for example, a sharp-crested weir 0.75 m. high, it was found that for a head over the crest of 0.2 m. the coefficient of discharge c was .433 for a free nappe. .46 " a depressed nappe. .497 " a drowned nappe. .554 " an adhering nappe. Beam Weirs. These weirs are formed of squared timbers laid one above the other to any required height, the weir faces being vertical and the crest, or sill, having a width e equal to that of the timbers. Free Nappe. The nappe may either spring clear from the up-stream edge, when the case becomes that of a sharp-crested weir, or it may remain in contact with the sill and spring clear from the down-stream edge. The first case is at once realized 4f H* exceeds 2 5pen channel with horizontal bed. Thickness of Nappe on Weir Crest. Let / = thickness of nappe. For a sharp-crested weir and free nappe - - varies from .85 to .86. //2 For a sharp-crested weir and drowned nappe 77 increases with - being .8 when - = .4, .855 when - = i, and .87 when > i. As the down-stream level rises -,- increases, exceeding .9 for the undulating condition, and necessarily tends to unity as the difference of level between the crest and the down-stream surface is greatly diminished. In beam weirs with free nappes - > .9 for small values of , and de- cieases as the head increases until the ratio becomes .855, when the nappe is on the point of separating from the sill. In beam weirs with drowned nappes the variation of is somewhat //2 complicated. The ratio diminishes until a minimum is reached, and io8 BERNOULLI'S THEOREM. then increases and approximates to values which are the same as in the* case of sharp-crested weirs. In wide-crested weirs with sloping faces is very variable. Gener- ally it increases as the down-stream slope diminishes, and diminishes with the up-stream slope. In weirs in which the crest is connected with the up-stream face by a curved surface may be less than .8, but the I , determination of the nappe thickness is in such case much less accurate. 28. Bernouilli's Theorem. A simple proof of this theorem is as follows: Consider an indefinitely small element of a stream-liife, of length ds and sectional area a. Let p, p + dp be the intensities of pressure at the ends. " TV be the specific weight of the fluid. " a. " " angle between the direction of motion of the element and the vertical. " dz be the vertical projection of ds^ so that dz = ds . cos a. Resolving in the direction of motion, ds cos a = accelerating force = mass x acceleration w . dv = a . ds . g dt ^ ii) dv = a . v . dt . pa (p x dp)a wa IV = av . dv. **' . *. a . dp iva . dz = av . dv, dp v . dv or dz -\ + - = o. w g Integrating, z + I - - + - = a const., is true for any fluid. If the fluid is water, w is constant, and then z + 2- + a const. EXAMPLES. 109 EXAMPLES. (N. B. In the following examples #- = 32 unless otherwise specified.) 1. 7" tons of water fall H feet per minute and are employed to turn turbines which transform into useful work orte half of the total energy x>f the water. What is the H.P. of the turbines ? TH ^ AHS. . ' 33 2. A turbine transforms into 9.72 H.P. of useful work the energy of the water fall ing jf feet from a Thomson V-notch in which the water stands at a constant level fcjtft. above the bottom of the notch. If the -coefficient of discharge is .o,*what is the efficiency of the turbine? Ans. .8. 3. A fall of 10 ft. supplies to a turbine 12 cu. ft. of water per sec. The turbine uses only 8 ft. of the fall, and the water leaves the turbine with a velocity of 8 ft. per sec. If 500 Ibs.-ft. are lost in frictional re- sistance, etc., find the efficiency of the turbine. Ans. .634.^ 4. 10,000 5o-volt incandescent and 250 45o-watt arc lamps are to be supplied with power from a waterfall having an effective head of 40 ft., ~2Q miles distant. Losses between lamps and converting apparatus at receiving end of transmission, 5$; ^efficiency of converting apparatus, 92$ ^line losses, 10$ ; ; losses in generators and transformers between line and turbine shaft, io^-f efficiency of turbine, 85^. Required, neces- sary flow of water per hour. Ans. 1,080,630 cu. ft. per hour. 5. A frictionless pipe gradually contracts from a 6-in. diameter at A to a 3-in. diameter at B, the rise from A to B being 2 ft. If the delivery is i cu. ft. per second, find the difference of pressure between the two points A and B. Ans. 504.6 Ibs. per sq. ft. 6. In a frictionless horizontal pipe discharging 10 cu. ft. of water per second, the diameter gradually changes from 4 in. at a point A to 6 in. at a point B. The pressure at the point B is 100 Ibs. per square inch ; find the pressure at the point A. Ans. 4118 Ibs. per sq. ft. 7. A ^-in. horizontal pipe is gradually reduced in diameter to in. -and then gradually expanded again to its mouth, where it is open to the atmosphere. Determine the maximum quantity of water which can be forced through the pipe (a) when the diameter of the mouth is ^ in., (b) when the diameter is f in. Also determine the corresponding velocities at the throat and the total heads (neglect friction, which, however, is very considerable). Ans. (a) .24 cu. ft. per min. ; 46.7 ft. per. sec. (b) .239 cu. ft. per min. ; 46.66 ft. per sec. no EXAMPLES. 8. A short horizontal pipe ABC connecting two reservoirs gradually contracts in diameter from i in. at A to in. at j#and then enlarges to. i in. again at C. If the height of the water in the reservoir over C be 12 ins., determine the maximum flow through the pipe and sketch the curve of pressures. Also obtain an equation for this curve, assuming the rates of contraction and expansion of the pipe to be equal and uniform. Ans. 4 cu. ft. per min. 9. In a diverging mouthpiece the diameter of the throat is .6 in.,, and the head of water over the axis is 30 ft. What is the discharge in. gallons per minute when the vacuum at the throat is 18.3 ins. of mer- cury ? Ans. 42. : "' 10. In a stream with still water 240 ft. above datum and flowing without friction, the velocity at a point 15 ft. above datum is 24 ft. per second. What is the pressure at this point? 9 'j r ~~ 6< - 37. A loo-gallon tank is 100 feet above ground and is filled by a i^- inch pipe connected with an accumulator having a 3-ft. cylr. piston loaded with 50 tons. If the mean lift of the piston is 10 ft. and if ^ of the head is lost in frictional resistance, how long will it t^.ke to fill the tank? Ans. 14.49 sees. 38. A bucket of water in a balance discharges 4 Ibs., of water per minute through an orifice in its base at 45 to the vertical, and is kept constantly full by a vertical stream which issues from an orifice 8 ft. above the surface with a velocity of 30 ft. per sec. Show that the bucket must be counterpoised by about .066 Ib. more than its weight. 39. The water in a vessel 9 ft. in height and 2 ft. in diameter is 8 ft. deep. In what time would one half of the water flow away through an orifice in the bottom i inch in diameter? If the orifice is closed and the vessel is made to rotate about its axis at. the rate of 76 T 4 T revolutions per minute, to what height will the water rise on the vessel's surface? If the orifice is opened, find velocity of efflux when the surface at the axis is 3 ft. above the orifice. Also find the difference of pressure-head in a horizontal plane 6 inches from the axis. Ans. 190.77 sees.; to the top ; 16 ft. per. sec.; 3 ins. (s 40. A cylindrical vessel, 10 ft. high and i ft. in diameter, is half full of water. Find the number of revolutions per minute which the vessel must make so that the water may just reach the top, the axis of revolu- tion being (i) coincident with the axis of the vessel, (2) a generating line of the vessel. Ans. (i) 483; (2) 241!. 41. A vessel full of water weighs 350 Ibs. and is raised vertically by means of a weight of 450 Ibs. Find the velocity of efflux through an orifice in the bottom, the head being 4 ft. and^ = 32.2. Ans. 17.02 ft. per sec. 42. A vessel full of water makes 100 revols. per min. Find the velocity of efflux through an orifice 2 ft. below the surface of the water at the centre, the diam. of the vessel being 3 ft. and c v = i. Ans. 33.4 ft. per sec. ./ What will be the velocity if the vessel is at rest? Ans. 11.3 ft. per sec. 114 EXAMPLES. 43. Show that when the water flowing over has a depth greater than .3874 ft. it is carried completely over the longitudinal opening, .83 ft. in width. At what depth does all the water flow in ? Ans. .221 ft. 44. A square box 2 ft. in length and 2 ft. across a diagonal is placed with a diagonal vertical and filled with water. How long will it take for the whole of the water to flow out through a hole at the bottom of .02 sq. ft. area ? (c = .625.) Ans. 97.52 sees. 45. A pyramid 2 ft. high, on a square base, is inverted and filled with water. Find the time in which the water will all run out through a hole of .02 sq. ft. at the apex. A side of the base is i ft. in length. (c = .625.) Ans. 5.656 sec 46. Find the discharge under a head of 25 ft. through a thin-lipped square orifice of i sq. in. sectional area, (a) when it has a border on one side, (b) when it has a border on two sides. Ans. (a) .3576 cu. ft. per sec.; (b) .3706 cu. ft. per sec. 47. A vessel in the form of a paraboloid of revolution has a depth of 16 in. and a diam. of 12 in. at the top. At the bottom is an orifice of i sq. in. sectional area. If water flows into the vessel at the rate of 2-^ cubic feet per minute, to what level will the water ultimately rise? How long will it take to rise (a) 1 1 in., (b) 1 1.9 in., (c) 1 1.99 in., (d) 12 in. above the orifice? If the supply is now stopped, how long (e) will it take to empty the vessel ? Ans. 12 inches; (a) 49.17 sec.; (b) 124.2 sec.; (c) 202 sec. ; (a) an infinite length of time ; (e) 11.3 sec. 48. If the vessel in Example 47 is a sphere i ft. in diameter, to what height will the water rise ? How long will it take for the water to rise (a} n in., (b) 12 in. above the orifice? Howjong (c) will it take to empty the vessel ? Ans. 12 inches ; (a] 67.16 sec.; (b) 81.46 sec.; (c) 24.13 sec. 49. In a vortical motion two circular filaments of radii r\ , r 2 , of ve- locities /i , s/a , and of equal weight W are made to change place. Show v * that a stable vortex is produced if = const.; and if r* > r\ , show that the surfaces of equal pressure are cones. fj 50. Sometimes the crest of a dam is raised by floating a stick L into the position Li , where it is supported against the verticals. The stick then falls of itself into position L 9 and rests on the crest. Explain the reason of this. 51. A 6-in. pipe discharges 8000 gals, per hour into a 9-in. pipe. Find the loss of head at the June- FIG. 78. tion. Ans. 1.58 ft. 52. Prove that for a Borda's mouthpiece running full the coefficient i of discharge is . EXAMPLES. 115 53. Find the discharge in pounds per minute through a Borda's rnouthpiece I in. in diameter, the lip being 12 in. below the water- surface, (a) when the jet springs clear from the edge, (b) when the mouth- piece runs full. Ans. (a) 81.845 \ (^) 1 15-74- 54. The surface of the water in a tank is kept at the same level ; obtain the discharge at 60 in. below the surface (a) through a circular orifice i sq. in. in area, (b) through a cylindrical ajutage of the same sectional area fitted to the outside, (c) through the same ajutage fitted to the inside, and determine the mechanical effect of the efflux in each case. Ans. (a] 4.85 Ibs. per sec.; 20.536 ft.-lbs. per sec. (b) 6.366 " " " ; 21.404 " (c) 5.49 " " " ; 13.725 " " " if running full. 3.69 " " " ; 16.638 " " " if jet springs clear. 55. Water is discharged under a head of 64 feet through a short cylin- drical mouthpiece 12 in. in diameter. Find (a) the loss of head due to shock, (b) the volume of discharge in cubic feet per second, (c) the energy of the issuing jet. (g = 32.) Ans. (a) 20.736 ft.; (b) 41.23 cub. ft. ; (c} 201.64 H.P. 56. If a bell-mouth is substituted for the mouthpiece in the preced- ing question, find the discharge and the mechanical effect of the jet. Ans. 49.28 cub. ft. per sec.; 344.2 H.P. 57. Compare the energies of a jet issuing under an effective head of loo ft. through (i) a 12-in. cylindrical ajutage, (2) a 12-in. divergent aju- tage, (3) a 12-in. convergent ajutage, the angle of convergence being 21. Draw the plane of charge in each case. Ans. (i) 393-8 H.P. ; (2) 672.28 H.P.; (3) 552.58 H.P. 58. Find the discharge through a rectangular opening 36 in. wide and 10 in. deep in the vertical face of a dam, the upper edge of the opening being 10 ft. below the water-surface. Ans. 40.2 cub. ft. per sec. 59. A centrifugal pump has a wheel of 2 ft. outside and i ft. inside diam., and also a large whirlpool chamber. Draw to scale a curve show- ing the pressure at different points in the wheel and whirlpool chamber when the water fills the pump but flows very slowly towards the point of discharge. Take i atm. as the pr. at the inlet surface. 60. A submerged sluice in the vertical face of a reservoir is 30 ft. wide. The effective head over the sluice is 18 inches. How high must the sluice be raised to give a delivery of 45,000 gal. per minute ? (c .6.) Ans. 8.164 1MS - 6r. The sill of a sluice in the vertical face of a reservoir is clear above' the tail-race ; the head of water above the sill is 5 feet. If the sluice is 24 ft. wide, what must be the opening to give 93,750 gals, per min.? {c = .6.) Ans. 12.3 ins. 62. A sluice in the vertical side of a reservoir is partially submerged,. n6 EXAMPLES. the surface of the tail-race water being 6 ins. above the sill. The surface on the upstream side is 2^ ft. above the sill. If the sluice is 18 ft. wide, what must be the total opening of the sluice to give 15,637^ tons (of 2000 Ibs.) per hour ? Ans. 1.203 ft*. c being .6. 63. Find the discharge in cub. ft. per sec. through a sharp-edge orifice, 6 ins. square, in a vertical plate, the centre of the orifice being 15 ins. below the water-surface, (a) if the velocity of approach is i ft. per second, (b) if the channel of approach is 3 ft. wide by 2 ft. deep. Ans. (a) i. 3478; (b) 1.34. 64. A reservoir half an acre in area with sides nearly vertical, so that it may be considered prismatic, receives a stream yielding 9 cub. ft. per second, and discharges through a sluice 4 ft. wide, which is raised 2 ft. Calculate the time required to lower the surface 5 ft., the head over the centre of the sluice when opened being 10 ft. Ans. 1079 sees. 65. Show that the energy of a jet issuing through a large rectangular orifice of breadth B is i2$B(If /A*), Hi , H* being the depths below the water-surface of the upper and lower edges of the orifice, and the coefficient of discharge being .625. 66. A reservoir at full water has a depth of 40 ft. over the centre of the discharging-sluice, which is rectangular and 24 in. wide by 18 in. deep. Find the discharge in cubic feet per second at that depth, and -afso when the water has fallen to 30, 20, and 10 ft., respectively ; find the mechanical effect of the efflux in each case, c being .625. Ans. 94.8 cu. ft.; 82.1 cu. ft.; 67 cu. ft. ; 47.4 cu. ft.; 431.2 H.P.; 280 H. P.; 152.5 H. P.; 53.95 H. P. 67. Require the head necessary to give 7.8 cu. ft. per second through iin orifice 36 sq. in. in sectional area, c being .625. Ans. 38.9 ft. 68. The upper and lower edges of a vertical rectangular orifice are <6 and 10 ft. below the surface of the water in a cistern, respectively; the width of the orifice is i ft. Find the discharge through it. Ans. 56.42 cu. ft. per sec. 69. The two sluices each 4 ft. wide by 2 ft. deep in a lock-gate are submerged one half their depth. The constant head of water above the -axis of the sluice is 12 ft. Find tlie discharge through the sluice, the velocity of approach being 4 ft. per second, c being .625. Ans. 16,626.2 cu. ft. per min. 70. Find the flow through a square opening, one diagonal being ver- tical and 12 in. in length, the upper extremity of the diagonal being .in the surface of the water, and c being .625. Ans. 1.724 cu. ft. per sec. ^71. To find the quantity of water conveyed away by a canal 3 ft. wide, a board with an orifice 2 ft. wide and i ft. deep is placed across the canal and dams it back until it attains a height of 2^ ft. above the bottom and if ft. above the lower edge of the orifice. Find the dis- charge in cubic feet per second, c being .625. Ans. 17.59, or 20.21 if orifice is drowned. EXAMPLES. 1 1 7 72. Six thousand gallons of water per minute are forced through a line of piping AC and are discharged into the atmosphere at C, which is 6 ft. vertically above A. The pipe AB is 6 in. in diameter and 12 ft. in length; the pipe BC is 12 in. in diameter and 12 ft. in length. Dis- regarding friction, find the "loss in shock" and draw the plane of charge. Ans. Loss of head in shock = 58.3 ft. 73. What quantity of water flows through the vertical aperture of a dam, its width being 36 in. and its depth 10 in. ; the upper edge of the aperture is 16 ft. below the surface. Ans. 50.65. cu. ft. per sec. 74. 264 cu. ft. of water are discharged through an orifice of 5 sq. ins. in 3 min. 10 sec. Find the mean velocity of efflux. Ans. 64 ft. per. sec. 75. One of the locks on the Lachine Canal has a superficial area of about 12,150 sq. ft., and the difference of level between the surfaces of the water in the lock and in the upper reach is 9 ft. Each leaf of the gates is supplied with one sluice, and the water is levelled up in 2 min. 48 sees. Determine the proper area of the sluice-opening. (Centre of sluice 20 ft. below surface of upper reach and c = .625.) Ans. Area of one sluice = 43.39 sq. ft. 76. The horizontal section of a lock-chamber may be assumed a rectangle, the length being 360 ft. When the chamber is full, the sur- face width between the side walls, which have each a batter of i in 12, is 45 ft. How long will it take to empty the lock through two sluices in the gates, each 8 ft. by 2 ft., the height of the water above the centre of the sluices being 13 ft. in the lock and 4 ft. in the canal on the down- stream side. Ans. 600.75 sec., c being .625. 77. Water approaches a rectangular opening 2 ft. wide with a velocity of 4 ft. per second. At the opening the head of water over the lower edge = 13 ft., and over the surface of the tail-race = 12 ft. ; the discharge through the opening is 70 cu. ft. per second. Find the height of the opening, c being .625. Ans. 1.022 ft. 78. The water in a regulating-chamber is 8 ft. below the level of the water in the canal and 8 ft. above the centre of the discharging-sluice. Determine the rise in the canal which will increase the discharge by 10 percent. Ans. 1.68 ft. The horizontal sectional area of the chamber is constant and equal to 400 sq. ft. ; in what time will the water in the chamber rise to the level of that in the canal, if the discharging-sluice is closed; the sluice be- tween the canal and chamber being 3 sq. ft. in area? Ans. 150.83 sec. 79. A lock on the Lachine Canal is 270 ft. long by 45 ft. wide and has a lift of Sf ft. ; there are two sluices in each leaf, each 8| ft. wide by 2^ ft. deep; the head over the horizontal centre line of the sluices is 19 ft. Find the time required to fill the lock. Ans. 163.5 sec - 80. The locks on the Montgomeryshire Canal are 81 ft long and 7f f*. wide; at one of the locks the lift is 7 ft. ; a 24-in. pipe leads the water Ii8 EXAMPLES. from the upper level and discharges below the surface of the lower level into the lock-chamber ; the mouth of the pipe is square, 2 ft. in the side, and gradually changes into a circular pipe 2 ft. in diameter.- Find time of filling the lock, (c = i.) Ans. 132.11 sec. 81. A canal lock is 115.1 ft. long and 30.44 ft. wide; the vertical depth from centre of sluice to lower reach is 1.0763 ft., the charge being 6.3945 ft.; the area of the two sluices is 2 x 6.766 sq. ft. Find the time of filling up to centre of sluices, (c = .625 for the sluice, but is reduced to .548 when both are opened.) Also, find time of filling up to level of upper reach from centre of sluice-doors. Ans. 25 sec. ; 298 sec. 82. How many gallons of water will flow through a 90 notch in 24 hours if the depth of the water is 27 ins. for the first 8 hours, 12 ins. for the second 8 hours, and 3 ins. for the third 8 hours, c being .6 ? Ans. 3,974,400. 83. Show that in a channel of V section an increment of 10 per cent in the depth will produce a corresponding increment of 5 percent in the velocity of flow and of 25 per cent in the discharge. 84. The angle of a triangular notch is 90. How high must the water rise in the notch so that the discharge may be 1000 gallons per minute? Ans. 12 ins. very nearly. 85. A reservoir, rectangular in plan and 10,000 sq. ft. in area, has in one side a 90 triangular notch 2 ft. deep. If the reservoir is full, in what time will the level sink 6 ins. ? Ans. 496.87 sees. 86. How long will it take to lower by 3 ft. the surface of a reservoir of 640,000 sq. ft. area through a 90 V notch 4 ft. deep ? Ans. 40.50 hrs., c being .6. 87. Find the discharge in cubic feet per second through a 90 notch when the depth of water in the notch is 4 ft., c being .617. Ans, 84.24. 88. A pond whose area is 12,000 sq. ft. has an overfall outlet 36 in. wide, which at the commencement of the discharge has a head of 2.8 ft. Find the time required to lower the surface 12 in. Ans. 354.58 sec. '89. How much water will flow through a rectangular notch 24111. wide, the surface of still water being '8 in. above the crest of the notch ? (Take into account side contraction.) Ans. 3.383 cu. ft. per sec. ' 9* A weir passes 6 cubic feet per second, and the head over the crest is-8 inches. Find the length of the weir, c being .625. Ans. 3.3068 ft. 91. A wein 400 ft. long, with a 9-111. depth of water on it, discharges through a lower weir 500 ft. long. Find the depth of water on the latter. Ans. .6457 ft. 92. A weir is 545 ft. long ; how high will the water rise over it when it rises .68 ft.' upon an upper weir 750 ft. long ? Ans. .8413 ft. ; 93. What should be the height of a drowned weir 400 ft. long, to deepen the water on the up-stream side by 50 per cent, the section of EXAMPLES. 119 the stream being 400 ft. x 8 ft., and the velocity of approach 3 ft. per second ? Ans. 7.084 ft. *94. The depth of water on the crest of a rectangular notch 5 ft. long is 2 feet. Find the discharge when the notch has (a] two end contrac- tions, (b) one end contraction, (c) no end contraction, c in each case being f . Ans. (a) 43.369 cu. ft. per sec.; (b} 45.254 cu. ft. per sec.; (c) 47.14 cu. ft. per sec. 95. Show that upon a weir 10 ft. long with 1 2 ins. depth of water flowing over, an error of TTTOT f a f ot m measuring the head will cause an error of 3 cu. ft. per minute in the discharge, and an error of T Q of a foot in measuring the length of the weir will cause an error of 2 cu. ft. in the discharge. 96. In the weir at Killaloe the total length is iioo ft., of which 779 ft. from the east abutment is level, while the remainder slopes i in 214, giving a total rise at the west abutment of 1.5 ft. Calculate the total discharge over the weir when the depth of water on the level part is 1.8 ft., which gives .3 ft. on highest part of weir. (Divide slope into 8 lengths of 40 ft. each, and assume them severally level, with a head equal to the arithmetic mean of the head at the beginning and end of each length.) Ans. 7496 cu. ft. per sec. 97. A watercourse is to be augmented by the streams and springs above its level. The latter are severally dammed up at suitable places and a narrow board is provided in which an opening 12 in. long by 6 In. deep is cut for an overfall ; it was surmised that this would be sufficient for the largest streams ; another piece attached to the former would reduce the length to 6 in. for smaller streams. Calculate the delivery by the following streams : In No. i stream with the 12-in. notch, depth over crest = .37 ft. " No. 2 " " " 6-in. " " " " = .41 ft. " No. 3 " " " 12-in. " " " = .29 ft. " No. 4 " " 6-in. " " " " =.19 ft. (Take into account the side contractions.) Ans, No. i, .696cu. ft. ; No. 2, .3658 cu. ft.; No. 3, .4904 cu. ft.; No. 4, .1275 cu. ft. 98. A rectangular notch has two complete end contractions and the length of the crest is three times the depth of the water on the crest. What must be the length of the crest to give a minimum discharge of 18,750 gals, per minute, c being f ? Ans. 5.87 ft. 99. A stream 30 ft. wide, 3 ft. deep, discharges 310 cu. ft. per second; a weir 2 ft. deep is built across the stream. Find increased depth of latter, (a) neglecting velocity of approach, (b) taking velocity of approach into account. Ans, (a) 1.26 ft. to 1.265 ft-J (^) I - I 9 ft- 100. In a stream 50 ft. wide and 4 ft. deep water flows at the rate of zoo ft. per minute ; find the height of a weir which will increase the depth 120 EXAMPLES. to 6 ft., (i) neglecting velocity of approach, (2) taking velocity of approach? into account. Ans. (i) 4.4126 ft.; (2) 4.4305 ft. 101. A stream 50 ft. wide and 4 ft. deep has a velocity of 3 ft. per second ; find the height of the weir which will double the depth, (i) neglecting velocity of approach, (2) taking velocity of approach into ac- count. Ans. (i) 5.651 ft.; (2). 5. 6862 ft. 102. A stream 80 ft. wide by 4 ft. deep discharges across a vertical section at the rate of 640 cu. ft. per second ; a weir is built in the stream, increasing its depth to 6 ft. Find the height of the weir. Ans. 4.233 ft. 103. Salmon-gaps are constructed in a weir; they are each 10 ft. wide and their crests are 18 in. below the weir crest. Calculate the discharge down three of these gaps, the water on the level part of the weir being 8 in. deep. Ans. 238.15 cu. ft. per sec. 104. A channel of rectangular section and 20 ft. wide conveys 3,600,000 gallons per hour, the depth of the stream being 8 ft. A dam 2 ft. high is built across the channel. Find the " height of swell " (a) dis- regarding the velocity of approach, (b) taking the velocity of approach into account. Ans. (a) .07 ft.; (b) .0545 ft. 105. The water in a flume 8 ft. wide is 3 ft. deep and is supplied from a sluice 6 ft. wide at the rate of 27,000 gals, per minute. If the coeffi- cient of contraction is unity and if 10 per cent is allowed for fractional loss, find the difference of level between the water-surfaces above the sluice and in the flume when the sluice opening is (a) i ft., (b) 2 ft. Ans. (a) 2.32 ft.; (b) .31 ft. 106. A stream of rectangular section 24 ft. wide delivers 145 cu. ft. per second. The edge of a drowned weir is 15 ins. below the surface of the water on the down-stream side. Determine the difference of level between the surfaces of the water on the up- and down-stream sides, the velocity of approach being 2 ft. per second. Ans. 7.9 ins. CHAPTER II. FLUID FRICTION AND PIPE FLOW. I. Fluid Friction. The term fluid friction is applied to the resistance to motion which is developed when a fluid flows over a solid surface, and is due to the viscosity of the fluid. This resistance is necessarily accompanied by a loss of energy caused by the production of eddies along the surface, and similar to the loss which occurs at an abrupt change of section, or at an angle in a pipe or channel. Froude's experiments on the resistance to the edgewise motion of planks in a fluid mass, the planks being T S F in. thick,. 19 in. deep, and I to 50 ft. long, each plank having a fine cutwater and run, are summarized in the following table: Length of Surface in Feet. Nature of Surface 2 Feet. 8 Feet. 20 Feet. 50 Feet. Covering. A B C A B C A B C A B C Varnish 2 .OO . 4 1 . 3QO i S* q2^ .26.1 i S^ 278 o in r 81 2Cf> Paraffine oft 070 I Q-l O T A 260 27 I Tinfoil "> 16 2QC I . QQ 278 26 "\ L yj I OO 262 "O/ 2J/1 r S-; ? 16 Calico 1-93 .87 725 1.92 .626 54 1.89 5.31 447 I.&7 47-1 .232- 423. Fine sand 2.OO .81 . 6go 2 . oo .583 450 2.00 .480 .38.4 2.O(J .40^ 337 Medium sand 2.0O .go 730 2.OO .625 .488 2.OO F.34 .46*; 2.0O .488 .456. Coarse sand 2.0O I. IO .880 2. CO 714 .520 2.OO .588 .490 Columns A give the power of the speed (v) to which the resistance is approximately proportional. Columns B give the mean resistance, in pounds per square 121 122 FLUID FRICTION. foot, of the whole surface of a board of the lengths stated in the table.' Columns C give the resistance, in pounds, of a square foot of surface at the distance sternward *V m the cutwater stated in the heading, each plank having a standard speed of 10 ft. per second. The resistance at other speeds can be easily cal- culated. An examination of the table shows that the mean resistance per square foot diminishes as the length of the plank increases. This may be explained by the supposition that the friction in the forward portion of the plank develops a force which drags the water along with the surface, so that the relative velocity of flow over the rear portion is diminished. Again, the -decrease of the mean resistance per square foot is .132 Ib. when the length of a varnished plank is increased from 2 to 20 ft. , while it is only .028 Ib. when the length increases from 20 to 50 ft. Hence for greater lengths than 50 ft. the decrease of resistance may be disregarded without much, if any, practical effect. Thus, generally speaking, these experiments indicate that the mean resistance is proportional to the nth power of the relative velocity, ;/ varying from 1.83 to 2.16, and its average value being very nearly 2. Colonel Beaufoy, as a result of experiments at Deptford, also assumed the mean resistance to be proportional to the ?/th power of the relative velocity, the value of n in three series of observations being 1.66, 1.71, and 1.9. The frictional resistance is evidently proportional to some function of the velocity, F(?'), which should vanish when v is nil, as when the surface is level, and should increase with v. Coulomb assumed the function / 7 (r) to be of the form av -\- biP, -a and b being coefficients to be determined by ex- periment. Experiment shows that when b does not exceed 5 ft. per minute the resistance is directly proportional to the velocity, but that ic is more nearly proportional to the square FLUID FRICTION. 123 of the velocity When the velocity exceeds 30 ft. per minute ; <5r, F(v). = av when sv ~ 5 ft., per minute, and F(v) = fo' 2 when ^ ~ 30 ft. per minute. Again, observations on the flow of water in town mains indicate that no difference of resistance is developed under widely varying pressures, and this independence of pressure is also verified by Coulomb's experiment showing that, if a disc is oscillated in water, there is no apparent change in the rate of decrease of the oscillations, whether the water is under atmospheric pressure or not. From the preceding and other similar experiments the fol- lowing general laws of fluid friction have been formulated: (1) The frictional resistance is independent of the pressure between the fluid and the surface over which it flows. (2) The frictional resistance is proportional to the are of the surface. (3) The frictional resistance is proportional to some func- tion, usually the square, of the velocity. To these three laws may be added a fourth, viz. : (4) The frictional resistance is proportional to the density and viscosity of the fluid. A fifth law, viz., that "the frictional resistance is inde- pendent of the nature of the surface against which the fluid flows," has been sometimes enunciated, and at very low velocities the law is approximately true. At high velocities, however, such as are common in engineering practice, the resistance has been shown by experiment, and especially by the experiments carried out by Darcy, to be very largely influenced by the nature of the surface. 124 FLUID FRICTION. Let p be the (fictional resistance in pounds per square foot of surface at a velocity of I ft. per second. Let A be the area of the surface in square feet. Let i' be the relative velocity of the surface and the water in which it is immersed. Let R be the total frictional resistance. Then from the laws of fluid friction R = p . Av\ <2, w being the specific weight of the fluid*, Then R =fwA . *g The coefficient f is approximately constant for any given surface, and is termed the coefficient of fluid friction. The power absorbed by the frictional resistance . commencing at 15.4 ft. (4.7 m.) from the inlet, i.e., at a, point where the pipe was running full and the flow was steady. The upper ends of the piezometers terminated on a vertical plank so placed as to allow the water-levels in them to be observed and compared. In any two consecutive piezometers the difference of level, which is of course constant, represents, the frictional loss of head in a i64-ft. length of pipe. From the results of these experiments Darcy made the following deductions : (a) The frictional resistance depends upon the material ancf condition of the pipe. For example, the resistance to flow is much less in a glass than in an iron pipe, and is approximately twice as great in DARCY'S EXPERIMENTAL RESULTS. 127 pipes which have become incrusted with use as in new clean pipes. It must be remembered, however, that although numer- ous experiments have been made with new pipes, there have been comparatively few experiments with old pipes. Thus, in pipes in which the velocity of flow exceeds 4 ins. per second, Darcy considered it more correct to express a ( = -} in the form \ g' I +* d being the diameter of the pipe, and a and ft coefficients to be determined by experiment. The following values for a and ft are given by Darcy: ft For drawn wrought-iron or smooth cast-iron pipes 00x31545 .000012973 For pipes with surface covered by light incrustations .0003093 .00002598 Without sensibly altering the values of these coefficients they can be put into the following simple form : ?;(+=> d being the diameter in feet, and ^ being .005 or .01 according as the pipes are clean or haVe become slightly incrusted. (b) The coefficient b is not constant, but varies slightly both with the diameter and the velocity ', its value diminishing as d or v increases. In practice it is assumed that b is constant and the error involved has the advantage of giving to the pipe a larger sec- tional area than is actually required for a given discharge. Thus allowance is partially made for the incrustations with which the surface gradually becomes covered. 128 DARCY'S EXPERIMENTAL RESULTS. Darcy proposed to include all cases in the more general form = +K + - ; -- in which, for new and smooth iron pipes, a = .00001350 a' = .000031635 ft = .000012402 ft' .00000016186 This value for is rarely if ever used. TABLE GIVING DARCY'S VALUES OF / FOR VELOCITIES EXCEEDING 4 IN. PER SECOND. Diam. Value of f. Diam. Value of/. Diam. Value of/. of of of Pipe Pipe Pipe in New Incrusted in New Incrusted in New Incrusted Inches. Pipes. Pipes. Inches. Pipes. Pipes. Inches. Pipes. Pipes. 2 .0075 .0150 Q .00556 .OIIII 27 .00519 .01037 3 .00667 01333 12 .00542 .01083 30 .00517 .01033 4 .00625 .0125 15 00533 .01067 36 .00514 .OIO28 5 .0060 .OI2 18 .00528 .01056 42 .OO5I2 .OIO24 6 .00583 .01167 21 .00524 .01048 4 8 .00510 .01021 7 .00571 .01.143 24 .00521 .OIO42 54 .00509 .01019 8 -00563 .OII25 Weisbach gives the formula /= .0036 ,00429 Poiseuille's experiments indicate that the surface friction in capillary tubes is directly proportional to the velocity, but in pipes, in ordinary practice, the frictional resistance is certainly more nearly proportional to the square of the velocity, and must be largely due to eddies which are the more readily formed as the viscosity diminishes. This viscosity, again, POISEUILLE'S EXPERIMENTS. 129 increases as the temperature falls, and the surface friction is diminished by about I per cent for every rise of 5 F. in the temperature. The resistance to the motion of a body in water, or to the flow of water along a surface, is evidently of two lands, the one due to surface contact, the other to the forma- tion of eddies. Hele Shaw's experiments clearly show the effect of surface contact upon stream-line motion and the manner in which the motion is modified by the presence of obstacles (Trans. Naval Architects, 1897-98), while the two kinds of resistance are plainly demonstrated by the interesting experiments of Osborne Reynolds. The water flows through FIG. 80. a glass pipe AB having a trumpet-shaped mouth A. A glass tube CD with a funnel E terminates in a pipette F y the axis of the pipette being in line with the axis of the pipe. The tube is filled with an aniline dye which is allowed to escape through the pipette in a thin thread-like stream, the discharge being governed by a small cock. So long as the velocity of flow in the pipe does not exceed a certain value, which Reynolds calls the critical velocity, the aniline thread is unbroken, so that the motion of the water is undisturbed and must be in parallel lines. As soon as the critical velocity is exceeded the colored thread is broken up, becoming sinuous in character, and the parallel stream-line motion is completely destroyed within a very short distance from the mouth of the pipe. According to Reynolds the critical velocity (z/ tf ), in metres per sec., is given by the formula 130 FLUID FRICTION. jj. being (?) for capillary tubes and ---= for ordinary pipes,, while -73 = i -j- .O336/ -[- .00022 it 2 , t being the temperature in degrees centigrade. It has been shown by H. T. Barnes, D.Sc., in his experi- ments on the specific heat of water, that, if water be heated while flowing through a tube at velocities less than the critical velocity, the temperature distribution in the column is not uniform. If the heat be applied electrically, by means of a wire threaded through the flow-tube, the hot water flows along the wire, leaving the walls of the tube almost entirely unheated. If the heat be applied to the walls of the tube, the colder water passes through the centre of the tube unheated, leaving a cloak of hot water along the sides. In neither case is there any tendency to mix as long as stream-line flow is maintained. A new method for determining the critical velocity of a fluid, based on the above experiments, has been recently worked out by Drs. Barnes and Coker in the McGill hydraulic laboratory. In this method, a sensitive mercury thermometer is placed exactly in the centre of a column of water as it emerges from the tube under examination, with the bulb just beyond the end. The walls of the tube are maintained at a constant temperature, slightly above that of the water flowing 'through, but for stream-line flow the temperature indicated by the thermometer will be that of the water in the head supplying the constant flow. The arrival of the critical velocity, at which stream-line flow becomes eddying and sinuous, is at once shown by a sudden small increase in the reading of the thermometer, and is due to the mixture of the water-film next the surface with the colder water flowing through the body of the pipe. The point is very sharply defined, and the method is in many cases far more applicable and convenient than the usual color- band test. SHIP RESISTANCE. 131 The experiments now in progress in the hydraulic labora- tory by Barnes and Coker are being made, both by the thermal and color-band methods, under the most favorable conditions for securing the perfectly steady conditions neces- sary for maintaining stream-line flow. The results so far obtained show that the effect of temperature is very marked in altering the point of instability of flow, and that this variation accords at least approximately with the formula quoted by Osborne Reynolds and taken from Poiseuille's experiments. The effect of pressure has been studied over a limited range,, and it has been shown that water flowing under a high head has greater stability, which means that there is a definite increase in the velocity at which stream-line motion breaks down. Indeed, under the present arrangements, it has been possible to maintain stream-line motion to very much higher velocities than is possible in experiments carried out with the apparatus used by Reynolds. 3. Resistance of Ships. The motion of a ship through water causes the production of waves and eddies, and the total resistance to the movement of a ship is made up of a frictional resistance, a wave-making resistance, and an eddy-making resistance. Although there is no theory 'by which the resist- ance at a given speed of a ship of definite design can be absolutely determined, Froude's experiments render it possible to make certain inferences and furnish some useful data. According to Froude, the frictional resistance' is sensibly the same as that of a rectangular surface moving with the same speed, of the same length as the ship in the direction of motion, and of an area equal to the immersed surface of the ship. Experiments seem to indicate that as the speed increases, the frictional resistance of well-designed ships with clean bottoms is from 90 to 60 per cent of the total resistance, and that the percentage is greater when the bottoms become foul. The wave-making resistance is especially affected by the form and proportions of the ship, depending, for a given 132 SHIP RESISTANCE. length, upon the proportions of the entrance, middle body, and run. For every ship there is a limit of speed below which the resistance is approximately proportional to the square of the speed, being chiefly due to friction, and beyond which it increases more rapidly than as the square. The eddy-resistance in the case of well-formed ships should not exceed about 10 per cent of the total resistance, and is often much less. Froude's law of resistance may be enunciated as follows: Let / t , / 2 be the lengths of a ship and its model. Let A^ , A^ be the displacements of a ship and its model. Let R^ , R 2 be the resistances of a ship and its model at the speeds ?/ t and v y Then, if the resistances are in the ratio of * _. _A1 R ~ A^~ I* Hence, too, the H.P. , and therefore ai so the coal consumption per hour, is proportional to Rv, that is, to .7 ,7 A* or /? or ?'', ^.nd the coal consumption per mile is proportional to A or / 3 or ?' 6 . Again, ^ is proportional to / 3 ; that is, to /X / 2 ; that is, to z/ 2 X ^f ; and it is sometimes convenient to express the resistance in pounds in the form R = k. PIPE FLOW ASSUMPTIONS. 133 u being the speed in knots, A the displacement in tons, and k a coefficient depending upon the type of ship and varying from .55 to .85 when the bottom is clean. Ex. If the New York, with a displacement of 10,000 tons and requir- ing 20.000 H. P. for a speed of 20 knots, is taken as the model for a new steamer which is to have a speed of 21 knots, then /2I\ e new steamer's displacement = 10,000 ( j 13,400, approximately, /2I\ 7 H. P. of new, steamer = 2o,ooof J = 28,000, approximately. 4. Pipe-flow Assumptions. In the ordinary theory of the flow of water in a pipe it is assumed that the water consists of thin plane layers perpendicular to the axis of the pipe, that each layer is driven through the pipe by the action of gravity and by the difference of pressure on its plane faces, and that the liquid molecules in any layer at any given moment will also be found in a plane layer after any interval of time. In such motion the internal work done in deforming a layer may be generally disregarded. It is further assumed that there is no variation of velocity over the surface of a layer, and this is equivalent to saying that each liquid molecule in a cross-section has the same mean velocity. The disagreement of these assumptions with the results of recent experimental researches will be referred to in a subse- quent article. 5. Steady Motion in a Pipe of Uniform Section. Since the motion is to be steady, the same volume Q cu. ft. of water will always arrive at any given cross-section of A sq. ft. with the same mean velocity v ft. per second. Then Q = Av. But since the pipe is of constant diameter, A is constant, and hence also v is constant, so that the mean velocity is the same throughout the whole length of the pipe. FLOW IN PIPE OF UNIFORM SECTION. Consider an elementary mass of the fluid A ABB, bounded by the pipe and by the two .cross-sections A A , BB. Let dt, be the length AB of the ele- ment, the length / ft. of the pipe being measured along the axis from any origin O. Let z, s -f- dz be the eleva- tions in feet above a datum line of the centres of pressure in the cross-sections AA, BB, respec- tively. Let/, p -{- dp be the intensi- ties of the pressures on these FIG. 81. cross-sections in pounds per square foot. Let P be the perimeter of the pipe. Let w be the specific weight of the water in pounds per cubic foot. Work Done by Gravity. In one second wQ Ibs. of water are transferred from A A to BB, falling through a vertical dis- tance of dz ft. Thus the work done by gravity per second a positive quantity if dz is negative, and vice versa. Work Done by Pressure. The total pressure on A A paral- lel to the axis = pA ; the total pressure on BB parallel to the axis = (p + dp)A. Therefore the total resultant pressure parallel to the axis in the direction of motion = A . dp, and the work done per second on the volume Q by this pressure = Q . dp. NOTE. The work done by the pressure at the pipe surface is nil, as Its direction is at right angles to the line of motion. Work Absorbed by Frictional Resistance. From the laws of fluid friction this work per second is evidently . = _ p . dl . F(v) X v = - -j- . Q . F(v) dl t FLOW IN PIPE OF UNIFORM SECTION. 135 the sign being negative as the work is done against a resist- ance. * *." . Since the motion is steady, the work done by the external forces must be equivalent to the work absorbed by the frictional esistance, and hence P wQ.dz Q . dp -^ Q . F(v) . dl = o, or *+* + 4;:S2U = a w A w Integrating, p ' P F(v) z -\- A- -r . - . / = a constant = H, 1 w ' A w so that /fft.-lbs. per pound of fluid is the uniformly distributed total constant energy. A -p is called the hydraulic mean radius of a pipe and will be denoted by in. Take W the value adopted in ordinary practice, /being the coefficient of friction. Then P fl v 2 z +w + ^ = Let z l , A l ,^> l be the elevation above datum, the area of the cross-section, and the intensity of the pressure at any point X on the axis of the pipe distant / x from the origin (Fig. 82). Let ,sr 2 , y4 2 ,/ 2 be the elevation above datum, the area of the cross-section, and the intensity of the pressure at any other point Y on the axis distant / 2 from the origin (Fig. 82). I3 6 FLOW IN PIPE OF UNIFORM SECTION. Then, from the equation just deduced, m 2g Hence -.+9- w m 2g f v\, fL -v* m m L being the length / 2 ^ of the pipe between the two points K FIG. 82. Let vertical tubes (pressure-columns) be inserted in the pipe at X and at K The water will rise in these tubes to the levels C and D, and evidently ^ being the intensity of the atmospheric pressure. FLOW IN PIPE OF UNIFORM SECTION. 137 Hence, if CX and DY are produced to meet the datum lirie in E and F, and Therefore = CE + w r w = DF + L. w w w \ m G being the point in which the horizontal through C meets FD produced. DG is called the "virtual fall " of the pipe, being the fall of level in the pressure-columns ; and since there would be no fall of level if the friction were nil, DG is said to be the head lost in friction in the distance XY. Denote this head by h ; then m zg'j and therefore h f v* L ~ ' m 2g* This ratio -j is designated the virtual slope of the pipe, and is the head lost in friction per unit of length. It will be denoted by 2, so that h. f V 2 L " " mT 2g* If the section of the pipe is a circle of diameter d, or a square with a side of length d, then _A d. ~P= 4 > " *- I3 8 INFLUENCE OF PIPE'S INCLINATION. and h . _ 4/ v* v^ L (f^ 2g r ' where a = and r is the radius. g 6. Influence of the Pipe's Position and Inclination on the Flow. In Fig. 82 join CD. Now since the fall of level (h} is proportional to L, the free surface in any other column between X and Y must also be on the line CD. Thus the pressure /' at any intermediate point M distant x (= XM) from X is given by - = MN + -- = CX+ -^(DY CX\ + ^. w w /> w Hence, at every point of a pipe laid below CD, the fluid pressure (/') exce'eds the atmospheric pressure (/ ) by an amount w . MN, so that if holes are made in such a pipe, the water will flow out and there will be no tendency on the part of the air to flow in. In pipes so placed vertical bends may be introduced, care being taken to provide for the removal of the air which may collect in the upper parts of the bends. If the line of the pipe coincides with CD, i.e., with the virtual slope or line of free surface level, MN =. o, and the fluid pressure is equal to that of the atmosphere. If holes are now made in the pipe, it can easily be shown by experiment that there will be neither any tendency on the part of the water to flow out nor on the part of the air to flow in. Next take CC' = DD' = ^, and join C'D'. w If the pipe is placed in any position between CD and C f D' t MN becomes negative, and the fluid pressure in the pipe is less than that of the atmosphere. If holes are made in this pipe^ there will be no tendency on the part of the water to flow out, but the air will flow in. Thus, if a pipe rises above the line PIPE FORMUL/E. 139 of virtual slope, there is a danger of air accumulating in the pipe and impeding, or perhaps wholly stopping, the flow. No vertical bends should be introduced, as the air is easily set free and would collect in the upper parts of the bends, with the effect of impeding the flow and of acting detrimentally upon the water itself, which the liberation of the air renders less wholesome. If the line of pipe coincides with C'D', then the fluid pressure is nil. Finally, if the pipe at any point rises above C'D' t the pressure becomes negative, which is impossible. In fact, the continuity of flow is destroyed, and the pipe will no longer run full bore. Air will be disengaged and will rise and collect at the point in question, so that in order to prevent the flow being wholly impeded, it will be necessary to introduce an air- chamber at this point from which the air can be removed when required. NOTE. In the preceding it has been assumed that the pipe is straight. If the pipe is curved, so also is the line of virtual slope. In ordinary practice, however, the vertical changes of level in a pipe at different points are small as compared with the length of the pipe, and distances measured along the pipe are sensibly proportional to distances measured along the horizontal projection of the pipe. Hence the line of virtual slope may be assumed to be a straight line without error of practical importance. 7. Formulae of Darcy, Hagen, Thrupp, Reynolds, etc. Darcy arranged the results of his experiments in a table drawn up as follows : Velocities in m./sec. pw A .10 .12 T 3 .14 to 3 m./sec. h }, h k D A L Q ~L Q L Q L Q. 140 PIPE FORMUL/E. The first column gives the several diameters. The second column gives the corresponding sectional areas. The remaining columns give the several velocities of flow from 4 ins. (. I m.) up to 10 ft. (3 m.) per second, and each velocity column is subdivided into two columns, the one giving the loss of head {-=?} per unit of length, and the other giving the discharge (Q). An examination of the table of Darcy's results shows that approximately the loss // is directly proportional to the length L of pipe under consideration and to the square of the velocity, z>, and is inversely proportional to the diameter d. Therefore L 4fL v 2 _v 2 where a = = i + , (p. 127.) In Hagen's formula, viz., h av" the values of a, n, and x vary with the velocity, the diameter, and with the roughness of the surface. The results obtained by this formula are in accord with the results of Pearsall's experiments with pipes in good condition and of diameters varying from .9 ft. to 4 ft., when a = .0004, n = 1.87, and x = 1.4, but the agreement is not so close if the pipe surface is very smooth. If the pipes are rough, the approximate values of the indices are a .0007, n = 2, and x i.i, PIPE FORMUL/E. 141 but these values must necessarily vary with every different class of pipe. Various modifications of Hagen's formula have been pro- posed, and perhaps one of the best is that contained in a paper by Thrupp, read before the Society of Engineers (London) in 1887. It may be written i L fm*\* = cosec. of slope angle -r = I k \ CV I ' 2 m m being substituted for x when m is small. The values of n, c, x, y, and z, for a pipe or channel, are given by the following table : Surface. n c X y z Wrought-iron pipes 1. 80 0.004787 0.65 0.018 0.07 Riveted sheet-iron pipes.... 1.825 0.005674 0.677 New cast-iron pipes -j 1.85 2.00 0.005347 0.006752 0.67 0.63 Lc3.d pines I 7< Orw>c o*>/i f * /D . uy ^ ,._-_$. O.O2 Pure cement rendering . .*. . j 1-74 i-95 0.004000 0.006429 0.67 0.61 Brickwork (smooth) 2. GO O.OO774.6 o 6 1 (roii| r h ) 2 OO r\ f\CkRR/% C. L LMJOO^S 0*02 5 0.01224 0.50 Unplaned plank .... 2 OO Q , o. 0004.5 1 o.oi 5 -3349 0.50 Small gravel in cement. . . . 2.OO 0.01181 0.66 0.03938 0.60 Large " " " 2.OO 0.01415 0.705 0.07590 I.OO Hammer-dressed masonry. 2.00 0.01117 0.66 0.07825 1. 00 Earth (no vegetation) 2.OO O OT C lf\ OT> Rough stony earth 2.00 vJvJl j^^J 0.02144 1 * 0.78 Osborne Reynolds has propounded a simple law of resist- ance embracing the results of Poiseuille and Darcy, and taking into account the effects of viscosity, temperature, etc. This law may be expressed in the form (the units being a foot and a second) i = the slope = = L 142 PIPE FORMUL/E. in which A = 1.917 X 10, B 36.8, and i + -O336/-J- .000221^ / being the temperature in degrees centigrade. Approxi- mately, the index n is I if the critical velocity is not ex- ceeded, and 1.7 to 2 for values of v greater than the critical velocity. According to Unwin the index of d is not exactly 3 n and should be determined independently. For a rough surface n = 2, for a smooth cast-iron pipe n = 1.9, and for a lead pipe n = 1.723 a limitation which is analogous to that found by Froude in his experiments upon surface friction. It may be noted that the sum of the exponents of v and d is constant and equal to 3. In a paper read before the Royal Society of New South Wales, 1897, Knibbs investigates the effects of temperature and records the results of a number of experiments, but the formula he deduces is too complicated to be of much practical value and requires further verification. Fournie has also studied temperature effect and has sug- gested a formula, but his results are not complete (Annales des Pouts et Chaussees, 1898). Again, a simple empirical law connecting v, m, and i may be expressed in the form in which c is a coefficient whose value is to be determined by g 32.16 -513 experiment. Taking c = _= -^ "->= 7-. then, if wf 62.42 X/ / 3// =y, this formula may be written z, = For values of;/ from .008 to .018 the results are practically the same as those obtained by substituting the same values for n in Kutter's more complicated formula (Chap. Ill) ; but while the PIPE FORMUL/E. 145 two formulae closely agree in ordinary cases, they both fail in extreme cases. The formula is also equally applicable to open channels (Chap. Ill), m being the mean hydraulic depth ; but Tutton has found that when m is small, and especially in the case of open channels, it is preferable to use the modified expression v = (Ii5_4 _ L \ ,/ ,-* \ n ml Lampe's well-known formula for iron pipes is v = 2O3.3iW/f > while Foss gives for the same case In 1867, M. Levy in his Theorie d'nn C our ant Liquide^, the units being a metre and second, gave : for new cast-iron pipes v = 3 6 . 4 j ri( i + Vr) } ; " cast-iron pipes in service v = 2O.${rt(i -\- 3 Vr)\%. To these Vallot added in 1888: for cleaned cast-iron pipes v = 32.5(^(1 -f- Vr)\*. The corresponding formulae, with a foot and seconds units, are: v 9$.24\mt(i + .7809 Vm)\^; v = 52.5i{w/(i + 2.3427 Vm)\l\ + .7809 Vw)|i Vallot also modified the expression for pipes in service, and deduced v = 64.7889*4 /^ in metric units, or v = 96. 27 w a 2 , a foot and second being the units* 144 GRAPHICAL REPRESENTATION OF PIPE FORMULA. Manning, in 1890, gave the formula 1.486 4 i v = m*i*, n n being the same as in Kutter's formula, Chap. III. Flamant, in 1892, deduced the expression B -| and gave the following values for c\ For tin pipe c 284. 5 " lead " c = 272.7 " glass " r = 262. i " wrought-iron and asphalted pipe c = 257.3 " new cast-iron and tarred pipe c 232.5 " lightly incrusted iron pipes in service., c = 205.4 8. Graphical Representation of the formula v = cm j i^. The preceding formulae are special applications of the general expression in which the coefficients c, x and y for any series of experi- ments can be graphically determined in the following manner: Taking logarithms, log v = log c + x log m + y log i ; and if /\ is a particular value of i corresponding to a value ^, of ', log z>j = log c + .r log w -)- ^ log z' r Then log ?' log v^y (log z log /\), and is the equation to a straight line, the rectangular coordi- nates being the logarithms of v and of i. Selecting any set of experiments and plotting the corresponding values of log v GRAPHICAL REPRESENTATION OF PIPE FORMULA. 145 1. .2 .4 .6 .8 FIG. 83. Flow through Wooden Pipes. 1. Hamilton Smith. Series 10 m = .0263 2. Darcy and Bazin. " 52*. m= .303 3. Darcy and Bazin. " 51 m = .505 4. Clarke, Moon Island conduit m = 1.50 4 series, 22 experiments. Formula; V = I29W 66 /- 51 . PIPE-FLOW DIAGRAMS. .e QLAS PIPE LOGS OF TO AND i 5. Darcy. 4. Darcy. 3. Smith. 2. Smith. .8 1 .2 .4 .6 .8 1. .2 .4 .6 FIG. 84. Flow through Glass Pipes. Series 1 1 ........................ ;;/ = .04075 i. Smith. 7 8 . m .01 ccc Q.. . m = .oiojx 5 series, 32 experiments. Formula : v = 141 to 169 PIPE-FLOW DIAGRAMS. NEW WROUGHT IRON AND ASPHALT COATED PIPE. .6 .8 2. .2 .4 .6 .8 1. .2 w* ^6 J3 FIG. 85. Flow through New Wrought-iron and Asphalt-coated Pipes. .4^.6 .8 3. $ A f Smith, ) } North Bloomfield f " Iben, Bonn pipe .... ;//; | Smith, [ "/ North Bloomfield J " Darcy. Series TO .. m: j Smith, / m . I North Bloomfield f Smith, Texas Creek, m Lampe m Tubbs, Rochester. . . m~ Iben, Sternchanze . . m: Smith, Humbug pipe ///: Smith, I Cherokee pipe f Tubbs, Rochester... m. Gale in Rowland, High Heads m- " " " m: m. 36 series, 195 experiments. Formula: For asphalt-coated v = 139 to 188 ;- 62 /-55. For new wrought-iron v = 127 to 165 w- 62 t-s$. I. Darcy. Series i. . . . tn = .OI 16. 2, Smith. " 6. ... m=. .01307 3- Smith. " 5.... tu- .021325 17- 4- Darcy. " 2.... rn .02182 18. 5- Smith. " 4. ?n = .0219 6. Darcy " 7 in .02198 19'. 7- Smith. i, 2.. m = .0219 7- Smith. " 3. ... in ~=^. .0218 2O. j Ehmann, ) 21. 8. ( Hahnwald j m .041 22. 9- Darcy. Series 3. ... m = .0324 22. 10. (Crozet, ) I Blue Ridge siphon f m .0625 23. 24, i i. Couplet , m . I '*'?'> ii. Iben, Deseniss St. .. i J 3" .08375 25- 12. Darcy. Series 8.. . . m = 06775 26. 13. Iben, Schoen St i = .12475 27. Ij. Iben. Series 5<* ..' . . m = .12475 28. 14. Ehmann, Stuttgart.. i = .1655 29. 15- Darcy. Series 9.. . . m = .16075 30. 1.22775 =.251 :.26 4 : -23375 = 3075 : -354 = 34325 : -75 i.o :.O2O8 :.O2O8 :.0208 PIPE-FLOW DIAGRAMS. 149 TUBERCULATED OR RUSTED PIPE OF IRON OR LIGHT MUD DEPOSITS. 1.2 LOGS. OF m AND i .2 .4 .6 .8 3. .2 .4 .6 .8 2. .2 .4 .6 .8 1. .2 .4 .6 .8 FIG. 86. Flow through Tuberculated or Rusted Pipe of Iron or Light Mud Deposits. 1. Iben, Koppel St., 19 years old. .. m 2. Iben, Schulweg, 19 years old m 3. Couplet m 4. Darcy. Series 12 m 5. Iben, Schulweg, 1 3 years old m 6. Fanning, rusted pipe m 7. Darcy. Series 14. .. m 8. Iben. " 1 $a, 22 years m 8. Iben, Strohhaus, 22 years m 9. Iben, Carolinen, 15 years m 10. Darcy. Series 19 m 1 1 . Couplet m 12. Iben, Rotherbaum m 12. Ehmann .... in 12. Iben, Heidenkampsweg, 25 years m 13. Iben, Hamm St. . . m 14. Iben, Glacis Chaussee. ;;/ 14. Duncan m 15. Bailey m = 08375 .1247 .0888 .02945 .1247 .020835 .0652 .2502-1- .2502 .2502 .1995 .266 .25 .2075 -h .4167 25 .250 .25 + .4167 PIPE-FLOW DIAGRAMS. 16. Leslie m = 17. Simpson. Series 3 m = 18. Leslie m 19. Couplet m 20. Simpson m = 21. Greene m 22. McElroy, Brooklyn main m = 23. Sherrerd, Pequannock main . . . , m = 23. Sherrerd, " " ;;/ = 30 series, 132 experiments. Formula : For light tuberculations v 87 to 132 r> For heavy tuberculations v = 31 to 80 w 3125 25 3333 4 3957 75 75 75 i.o >y // ^ -.-%! .2 2. .8 .8 .4 2 1. .5 .0 ' .4 .2 .8 .6 T.I ) H ^ / X 7 ^ ^ / ^ $s ^ oV /^ /s /> h > * X y^ / _^ >oV X" / /^ / // L^ k^ 1 " *x- / ^^ " > r.8 ^-^ ^ ^ ^ ^ ^ *r ? ^ ^ ^/ x^ J FITZGERALD'S EXPERIMENTS ON THE ROSEMARY PIPE. XS ^ ^ ^ Jo^' <^ > ^ -OGS . Of mM iD i r. .2 .4 .6 .8 3. .2 .4 .6 .8 2". .2 .4 .6 .8 F. '.2 \4 .6 .8 ( FIG. 87. Fitzgerald's Experiments on the Rosemary Pipe. 1. North pipe. Cleaned, asphalted 2. Both pipes. Tuberculated 4 series, 57 experiments. Formula : Cleaned, asphalted, v Tuberculated, v m i.oo m = i.oo PIPE-FLOW DIAGRAMS. X NEW CAST-IRON AND CEMENT LINED PIPES. .6 1.4 AtfID i 4. .2 .4 .6 .8 3. .2 .4 .G .8 2. 3, .4 .6 .8 1. .2 ..4 .6 .8 FIG. 88. Flow through New Cast-iron and Cement-lined Pipes. 2. 3- 4- 5- 6. 6. 7- 8. 8. 9- 10. Ehmann, Neckar St... m= .0827 + Darcy. Series 16 m .067175 Iben, Wenden St m= .0835 Iben, Haller St m= .1245 Darcy. Series 17 m .11237 Darcy. " 1 8 m = . 1 542 Ehmann, Stuttgart m - .20725 + Russell, St. Louis m = .25 Darcy. Series 22 m .4101 Fanning, cement-lined m= .41674- Friend, Seville m -4375 Woods, Newton (doubtful) m .5 Stearns, Rosemary pipe m = I .o 13 series, 79 experiments. Formula : v 126 to 158 w 66 /-? 1 . PIPE-FLOW DIAGRAMS. OLD, NEW AND CLEANED CAST IRON PIPE. .8 FIG. 89. Flow through Old, New, and Cleaned Cast-iron Pipes. 1. t)arcy. Series 13 ____ .................... ;;/ .02985 2. Meunier, Torcy, 28-30 years old ............ ;// = .1107 3. Darcy. Series 15 ........... ( . . . .......... m =. .0657 4. Meunier, Nogent sur Seine, new ............ m = .1035 5. Coffin, Taunton main, i\ years old ...... , , . m = .625 6. Meunier, Charenton, 2 years old ........... m = .1640 7. Darcy. Series 20 ................... .... m = .2007 8. Darcy. " 21 ................ ......... m =' .2436 9. Forbes, Brookline main, 8 years old ........ m = -3333 10. Humblot, i series, 10 years old ............. in = .4921 11. Meunier, Bercy .......................... m= .4921 12. Humblot, 3 series, 6, 7, and 12 years old . . . m .6562 13. Meunier, Canal de 1'Oise, i year old ...... m = .7382 14. Bruce, Blane Valley, new ............... m = i.o 16 series, 80 experiments. Formula: v = 96 to 148 w 66 /^ 1 . TABLES OF PIPE COEFFICIENTS. 15$ 10. Values of c, x, and y in the Formula v cm a5 i* / . Tutton found (see Reynolds' formula) that, in the general formula v cm x i y , x -\- y = a constant = 1.17, and therefore The values of c and y he has tabulated as follows: c y For tin pipe 1 89 .58 For lead pipe 168 .58 Older experiments give c 189. For brass, zinc, and glass pipe 165 .56 In one set of glass experi- ments c = 141. For wrought-iron pipe 160 .55 c varies from 127 to 165, ap- proximating to the higher number. For wood-stave pipe 125 .51 For new cast-iron or tarred pipe 130 .51 In tarred pipes r varies from 115 to 152, the values be ing about the same as in cast-iron pipes of same size. Benzinger gives for a 6o-in. cast-iron pipe c = 129. For pipe in service 104 .51 Generally c is about 105. In the Rosemary pipe c = 117. For tuberculated pipe 30 to 80 .51 For lap-riveted pipe 115 .51 c varies from 12510 135 for new to 1 10 to 1 14 for pipe in service. For rubber and leather hose. . 160 .51 For wrought-iron pipe asphalt- coated 170 .55 In some cases c = 140, and in the 48-in. pipe c = 199. For large brick conduits 129 .52 Unobstructed by shafts. For large brick conduits 91 .52 Fullerton Avenue conduit of Chicago water-supply. For large brick conduits 1 10 .52 Chicago land tunnel. 354 EXAMPLES. The values of c in this table are mean values and neces- sarily vary with age and roughness. Throughout the analysis of these experiments the total head was diminished by the loss of head at entrance, and in the cases in which this loss had not been found by means of piezometers it has been calculated from , c c being the C c 2 <" coefficient of contraction. Assuming forj/ the approximate value .5, Tutton's formula becomes Ex. i. The head over the sharp-edge entrance into a pipe, 1000 ft. long and passing i cu. ft. of water per sec., is 9 ft. Find the diameter, taking/ = .0055. -^Ylj 4 x .005$ x loooN _ 49 /, 22 ~ 64^2 "* //* ~) ~ 16. 4 ' For a first approximation, disregarding the first term on the right- hand side, which is small as compared with the second term, 49 22 _ 49 j_ ~ 16. 121 d* ~ 88 d* and d = .57319 ft. For a second approximation _22_\ 9 = - = 1.5, I6.I2U/ 4 V .573I9/ or d* = ^ x 39.8817, 9. 16. 121 and d .5787 ft. Ex. 2. The effective height of the grade line above the entrance into a clean iron 3-in. branch, 1000 ft. long, is 20 ft. 5 ins. How many peo- ple will the branch supply with 20 gallons of water per head per day of 24 hours ? / == .005(1 + L ^ = _L, >V 12 x I) 150' EXAMPLES. 155 and . v = 3i ft per sec. The delivery in cu. ft. per sec. - 7 4W ~ 64 The delivery in gallons per day _ II x 6i x 60 x 60 x 24 = 92,812^, 64 and the number of people served per day = - ' ? = 4640!, 20 or 4640. Ex. 3. Find the proper diameter of a rough pipe to give 60,000,000 of gallons every 24 hours^the slope of the pipe being i in 800. 22 d~ 60,000,000 Ir- or 6J . 60 . 60 . 24' 14000 99 h av n i Using Hagen s formula, viz., = = - , L d x 800 and taking a .0007, = 2, and .r = i.i, JL _'- 00 7 a .0007 / 14000 y ~ ; , /j are the intensities of pressure at the outer and inner surfaces; f is the intensity of stress at the radius r ; r Q , r l are the radii of the outer and inner surfaces. (See Appendix, Bovey's "Theory of Structures. ") Ex. i. An accumulator supplies a pressure of 700 Ibs. per sq. in. What length of 8-in. pipe will deliver 200 H.P. of useful energy with a loss of 20 per cent ? 250 H.P. enter the pipe. Therefore, if Q is the delivery in cu. ft. of water per sec., =<=-. 252 7 4\3 and V = ~- ft. per sec. Take / = .001; i + I = / , for a clean iron pipe. \ 12 X -/ rAi-vrv Then x 250 = loss = 62! . - . ioo 252 f 04 550 L being the length of the pipe. Therefore L 78,293.7 ft. = 14.8 miles. Ex. 2. The^ efficiency of an engine is .6; it burns 2 Ibs. of coal per hour per H.'P.^'and works 16 hours a day for 300 days in the year. The 160 PRESSURE DUE TO SHOCK. cost of the engine is $12 per H.P., and the cost of the coal $3 per ton. An amount of 4500 gallons of water per minute is to be raised a vertical height of 200 ft. What must be the minimum diam., D, of the pipe, assuming that the cost of the piping is D per lineal foot, and that / = .0064 ? Let h feet be the frictional loss of head. 41500 22 Z) 2 rhen ' smce =-~ v = 12 ' _ 4 x .0064 x 200 i_ (Y T 8 -) _ 2 /i68\ 2 i D ~ 64 " ~7J* ~ 2^ \ir) 'W' Again, let N be the number of H.P. Then N = - * 2 ' ^(200 + //) :> 55 25 j 2 /r68\ 2 i = - ] 200 + -- y,r n( 25 \. ii I D* Cost of coal capitalized at 5^ = N. 2 ' ,,^ -' ~ ~- = $288^". Cost of engine = f>i2A r . Cost of piping = $20o/>. Total prime cost = 3ooA T + 2ooZ^ ( A68V i ) = 3 00 X f| | 200 + {] W [ + 200A which must be a minimum. Therefore O 300 x f \ f -J + 200, and D = 3.98 ft. Hence, a!so, k = -* = .0,868 ft. No. of H.P. = N = 454 . 59. Capital cost = $137,173. 12. Pressure Due to Shock. Water flows through a line of piping with a velocity of v ft. per second, and at a certain point the motion is suddenly arrested by the closing of a valve, developing a sudden increase in the pressure at the valve of PRESSURE DUE TO SHOCK. 161 f Ibs. per sq. in. Water being slightly compressible losing T V*f its bulk under a pressure of 2 tons (of 2240 Ibs.) per square inch a compression-wave starts from the valve and moves backwards throughout the whole length L ft. of the moving column of water. The water still enters the pipe for the period of / seconds, during which the compression con- tinues. Let a ft. be the sectional area of the water-column ; " x ft. be the diminution in the length L of the wafer column ; ' ' K be the modulus of cubic elasticity of water = 300,000 Ibs. per sq. in. Then * L L~~ 1C x vt w and 1440 ft. = momentum of the fluid mass = aLv. o Hence i w L i w L i w L 2 f ~ 144 J 7 2 ' ~ 144 - ~P* ~ 144 ^~ "? J?' and - = velocity of the wave-propagation = /*/__. t \ w Substituting the values of g, K, and w, the velocity of wave- propagation is found to be about 4/20 ft. per second, which is also the velocity of sound in water. Ex. A volume of water 50 ft. in length, flowing through a pipe with a velocity of 24 ft. per sec., is quickly and uniformly stopped in one tenth of a second by closing a stop-valve. Find the increase of pressure per sq. in. in the pipe near the valve. The pres. per sq. in. = . - . * = 162.76 Ibs. 32 144 1 62 FLOW IN PIPE CONNECTING TWO RESERVOIRS. 13. Flow in a Pipe of Uniform Section and of Length Z, connecting two Reservoirs at Different Levels. Let z ft. be the difference of level between the water-surface in the two reservoirs. FIG. 90. The work done per second is evidently equal to the work done by the fall of wQ Ibs. of water through the vertical dis- tance z, and is expended (1) In producing the velocity of flow v ft. per second, which requires a head of z^ ft. and an expenditure of wQz l ft. -Ibs. of work per second; (2) In overcoming the resistance at the entrance from the upper reservoir into the pipe, which requires a head of -s-g ft. and an expenditure of wQz z ft. -Ibs. of work per second; (3) In overcoming the fractional resistance, which requires a head of z z ft. and an expenditure of wQz^ ft. -Ibs. of work per second. Thus wQz wQz l + wQz 2 + wQz^ , or Now z l ft., and the corresponding energy wQz l is ultimately wasted in producing eddy motions, etc., in the lower reservoir. v* # 2 may be expressed in the form n ft. , n being a coeffi- CHEZY'S LONG-PIPE FORMULA. cient whose value varies with the nature of the construction of the'entrance into the pipe. If the pipe-entrance is bell-mouth in form, n = .01 or .02, but if it is cylindrical, n = .49. Finally, _ ft -- it. r it. , m iv d 2g takine = f as is usual in practice. Hence W 2 T since Q = V, and/g'is assumed to be 32. For given values of Q and z a first approximate value of d may be obtained from the last equation by neglecting the term _|_ //). Call this value d v , and substitute it for the d in the term 7- within the brackets. A second approximation d may now be made by deducing d from the formula and the operation may be again repeated if desired. Generally speaking, I + n is usually very small as com- pared with ~ , , and maybe disregarded without error of prac- tical importance. The formula then becomes 4fL v 2 d 2g' which is known as Chezy's formula for long pipes. 1 64 LOSSES OF HEAD. The term I -j- ;/ need only be taken into account in the case of short pipes and high velocities. Ex. The difference of level between the water-surfaces of two reser- voirs, connected by a 24-in. pipe 6J miles in length, is 172^ ft. The pipe, having been in use for some time, has its inside surface coated with a deposit, and no special provision is made to diminish the resist- ance at the upper end. Determine the discharge into the lower reser- voir in gallons per hour. / = .01 ( Take = .01 i + Then .7* and v = 4 ft. per sec. Therefore the discharge in cu. ft. per hour and the discharge in gallons per hour = 45.2571- x 6i = 282,857}. 14. Losses of Head due to Abrupt Changes of Section, Elbows, Valves, etc. When the velocity, or the direction of motion of a mass of water flowing through a pipe, is abruptly changed, the water is broken up into eddies or irregular motions which are soon destroyed by viscosity, the correspondr ing energy being wasted. CASE I. Loss due to a sudden contraction. (Art. 17, Chap. I.) FIG. 91. FIG. 02. (a] Let water flow from a pipe (Fig. 91), or from a reser- voir (Fig. 92) into a pipe of sectional area A. LOSSES DUE TO SUDDEN CONTRACTION, ETC. 165 Let c c be the coefficient of contraction. * Then the area of the contracted section c c A , and I Iv the loss of head = I -- v } 2\c I 2gc c v where m = I The value of m has not been determined with any great degree of accuracy; but if c c = .64, then m .316. The value of c c is sometimes obtained from the formula 2.618- When the water enters a cylindrical (not bell-mouthed) pipe from a large reservoir, the value of _, m is about . 55- (b) Let the water flow across the abrupt charfge of section through a central FIG. 93. orifice in a diaphragm placed as in Fig. 93. Let-0 be the area of the orifice. Then c 2 v> = = ( - i) = m , (A \ 2 where m I I J . / 1 66 LOSSES DUE TO SUDDEN CONTRACTION, ETC. According to Weisbach, A ~ . i . 2 -3 4 5 c c = .616 .614 .612 .610 .607 m = 23I-7 50.99 19.78 9.612 5.256 a .6 7 .8 . -9 1. 00 c c = .605 .603 .601 .598 .596 m = 3-077 1.876 1.169 734 .48 (V) A diaphragm with a central r orifice of area a, olaced in a cylindri- \ cal pipe of sectional area J A as in ^jjjfif^^ 10 ^ Fig. 94. FIG. 94- The * * contracted area ' ' of the water = c c a and i IvA V ^ I A V the loss of head = I v \ = I 1 1 2g\c c a 1 2g\c f a I v* I A \ 2 where / = (^ ij . Generally m must be determined by experiment, but Weis bach gives the following results : 5.- .1 .2 3 4 5 m = .624 225.9 .632 47-77 643 30.83 .659 7.801 .681 3-753 *,= -712 *0 = 1.796 .7 -755 797 .8 .813 .29 9 .892 .06 i. oo 1. 00 oo LOSSES DUE TO ABRUPT ENLARGEMENT, ELBOWS, ETC. 167 CASE II. Loss due to a Sudden Enlargement (Fig. 95.) Let A^ = external area of small pipe. " A 2 = . " " " large 4< z /A loss of head = I / 2 \ v / 2 \ = - (-^ - v) := - (- - ij where m = l^r- NOTE. The losses of head in Case I (a) and in Case II may be avoided by substituting a gradual and regular change of section for the abrupt changes. CASE III. Loss of Head due to Elbows. (Fig. 96.) The loss of head due to the disturbance caused by an elbow is ex- pressed by Weisbach in the form m where m = .9457 sin 2 -f 2.047 sin4 ~> being the elbow angle. Weisbach deduced this formula from the results of experi- ments with pipes 1.2 in. in diameter. The velocity v^ with which the water flows along the length AB may be resolved into a component v with which the water flows along BC and a component u at right angles to the direction of v. The component u and therefore the corre- u 2 spending head, viz., - , is wasted. The component u evi- dently diminishes with the angle and becomes nil when a 1 68 LOSSES DUE TO ELBOWS, BENDS, ETC. gradually and continuously curved bend is substituted for the elbow. CASE IV. Weisbach gives the following empirical formula for the loss of head at a bend in a pipe, being the angle of curvature : h =, * where = .131 + ( d\\ "1. 847(^1 for a circular pipe of diameter d, p being the radius of curvature of the bend, and FIG. 97. for a pipe of rectangular section, $ being the length of a side of the section parallel to the radius of curvature (p) of the bend. According to Navier, ^ = (.0128 + .0186^, <$ R being the radius and L the length of the bend measured along the axis. As a result of recent experiments by Gardner S. Williams and others (Proc. Am. Soc. C. E., May, 1901) it is claimed that, down to a limit of 2j diameters, curves of short radius offer less resistance to flow than do curves of longer radius, which is contrary to the ordinary hypothesis. LOSSES DUE TO SLUICES, VALVES, ETC. 169 CASE V. Valves, Cocks, Sluices, etc. The loss of head in each of the cases represented by the several figures may be traced to a contraction of the stream similar to the contraction which occurs in the case of an abrupt change of section. The 2,2 loss may be expressed in the form m , and the following o tables give the results obtained by Weisbach : (a) Sluice in Pipe of Rectangular Section. (Fig. 98.) Area of pipe = a; area of sluice s. .9 .8 .7 .6 .5 .4 .3 .2 .1 m .00 .09 .39 .95 2.08 4.02 8.12 17.8 44.5 193 FIG. 98. (ft) Sluice in Cylindrical Pipe. (Fig. 99.) s ratio of height of opening to diameter of pipe. s= i .875 .75 .625 .5 .375 .25 .125 m = .00 .07 .26 .81 2.06 5.52 17.00 97.8 (c) Cock in Cylindrical Pipe (Fig. 100). s = ratio of cross-sections; 6 = angle through which cock is turned. FIG. 99. FIG. 101. 15 20 25 30 35 .772 .692 .613 .535 .458 75 1.56 45 315 50 25 55 .19 3.1 5-47 9-68 60 65 82 .137 .091 oo 31.2 52.6 106 206 486 CO LOSSES DUE TO CHANGES OF SECTION, ETC. (d) Throttle-valve in Cylindrical Pipe (Fig. 101). 6 = angle through which valve is turned. . If0= 5 10 15 20 25 30 35 40 m .24 .52 .90 1.54 2.51 3.91 6.22 10.8 If0= 45 50 55 60 65 70 90 m=iS.? 32.6 58.8 118 256 751 oo CASE VI. The fall of free surface-level, or loss of head, due to sudden changes of section, frictional resistance, etc., may- be graphically represented as in Fig. 102. -* FIG. 102. Let a length of piping AE connect two reservoirs, and let // be the difference of surface-level of the water in the reser- voirs. Let L l , r l be length and radius of portion AB of pipe. " L 2 , r 2 " " " " " " BC " " '' Z 4 , r 4 " " " " " " DE " " <4 u l , ? , 3 , 4 be the velocities of flow in AB, BC, CD, DE, respectively. LOSSES DUE TO CHANGES OF SECTION, ETC. 171 The reservoir opens abruptly into the pipe at A . '* There is an abrupt change at B from a pipe of radius r l to one of radius r 2 . There is an abrupt change at C from a pipe of radius r. 2 to one of radius r v o At D the water flows through an orifice of area A in a diaphragm. At E the velocity of the water as it enters the lower reservoir is immediately dissipated in eddies or vortices. Draw the horizontal plane amnop at a distance from the water-surface in the upper reservoir equal to the head due to atmospheric pressure. Draw vertical lines at A , B, C, D, E. Take u a ab =loss of head at the entrance A = .49 ; qc = " " " due to friction from A to B 2 ^ 1 ^-L l ; ' cd= " " " due to change of section at B( T ~ I) : W /.V r* = " " " due to friction from B to 7 = 2 l U -*-L^ ef= " " " due to change of section at 7=. 316 ; 2 S sg= u a due to f r i c tion from C to D =^~ . U ^-L- r* ig gh= " " " due to change of section at D= ( 7 1) \c e A I 2g tk = " " " due to friction from D to =^--L 4 ; u 2 ^/= " " corresponding to w = . I7 2 LOSSES DUE TO CHANGES OF SECTION, ETC. Through / draw a horizontal plane Ix. This plane must evidently be at a distance from the water-surface in the lower reservoir equal to the pressure-head due to the atmosphere. Then the total loss of head = lp = ab + cd+ef + gh + kl + qc + re + sg+tk, . , t , 2g r r^ 2g r r 3 2g r, 2g -49 ,/^ \'J , ^6 , /^!_ .\ J 1 , i I r7+ U?~' J I-;*"*' r s 4 ' h l^ ' V ^' + ^7 The broken line abcdefghkl is the hydraulic gradient. Ex. A clean 6-in. pipe, ^400 ft. long, containing a 60 bend with a i2-in. radius, a 90 bend with a ;2-in. radius, and a 120 bend with a 48-in. radius, discharges i cu. ft. of water per sec. into a clean 12-in. pipe, 200 ft. long, which again discharges into a clean 4-in. pipe, 500 ft long, containing four sharp knees, viz., one of 60, one of 90, one of 120, and one of 150. Find the total head wasted at the pipe entrance, at the bends, knees, sudden changes of section, and in the straight lengths. Let s/i, 77 2 , 2> 3 be the velocities of flow in the first, second, and third lengths, respectively. Then 22 I /I \ f 22 I 22 I /I \ 2 -- - } Vi = I = - (I 2 )77 2 = - -- ]V 7 4 \2/ 7 4 V ' 7 4\3/ EXAMPLES. 173 and 56 , 14 , 126 , * v\ = ft. per sec., V* = ft. per sec., 7/3 = ft. per sec. ii ii ii Head wasted at pipe entrance = - f ) .20332 ft. V* The head wasted at a bend = m b ^-^ , 1 80 ig where m b = .131 + 1.847^ J*. 2/J 26 4 ' d 6 i 2/J ~~ 144 ~" 24' m b = .14544 ; m b .13102727; d 6 i zp ~96 ~ 6' 1)l b .131113. Hence head wasted at 60 bend = .14544 x 3 \o x ^ x (ff) 2 .019632 ft.,. 90 " = .130273 XyVir x -fa x (H) a = - 0265303 ft., 120 " = .131113 x jfS x ,V x (ff) 2 = .035396 ft., and the head wasted in bends .081558 ft. The head wasted at a knee = nik , where m k -9457 sin 2 -- + 2.047 sin* . For a 60 knee (p = 120, mk = 1.8607 90 " 90, nik = .9846 " 1 20 " ... = 60, nik = .36436 1 50 " = 30, mk = .07254 Then head wasted at 60 knee = 1.8607 x ^(W) a .= 3- 8l 463 ft., " " 90 " = .9846 x ^V O^ 6 ) 2 = 2.01853'" " " I2 " = .36436 x ^VGW) 2 == -74^97 " I50 " = .07254 X TsH-TT 6 ) 2 = -14871 " and the head wasted in knees = 6.72884 ft. Head wasted at junction between 6-in. and i2-in. pipes ^V (ff if) 2 = .22778 ft., " 12-in. and 4-in. pipes = '-( ^J = .64783 ft. 64 \ 1 1 / and the head wasted at sudden changes of section = .87561 ft. 274 NOZZLES. For straight lengths take/ = .005(1 + 7 ^ r ^ = '-^i for 6-in. pipe, 12-in. " / i \ .065 / = .005 i H - " \ 12 X l) 12 Then head wasted 35 4 x -fUoo i in ist length =- -in 1 77 = 7-5592 ft., .065 4 x ^200 . . 12 I /H\ 2 . 2d " -z-l^ =7.01929 ft., i 64\ny .025 4 x ; and the frictional loss of head = 91.14329 ft. Hence the total head wasted = .081558 -f 6.72884 + .87561 -f 91.45729 = 99.1433 ft. 16. Nozzles. Let a pipe AB y of length / and diameter d, lead from a reservoir h ft. above the end B, Fig. 103. First. Let the pipe be open to the atmosphere at B. Then h head to overcome resistance to entrance / 7' at A (= n \ * Ci>^ \ = m J -f- head to overcome frictional resistance U= -~ -- -) > d igl NOZZLES. '75 + head corresponding to the velocity v in the pipe and (v* 1 \ = FIG. 103. Hence the height to which the water is capable of rising at B or, again, is = ~~ = h [n4- m + ~-V - - Second. Let a nozzle be fitted on the pipe at B. Let F be the velocity with which the water leaves the nozzle. Let D be the diameter of the nozzle-outlet. This diameter is very small as compared with the diameter d of the pipe. But and therefore _ v ' V. 4 4 ' so that V is very large as compared with v. 1 76 NOZZLES Also, li = head to overcome the resistance to entrance at A + head to overcome the resistance due to bends, etc. -f- head to overcome the frictional resistance in pipe -\- head to overcome the frictional resistance in nozzle i ' v '\ \ m - V 2f) -(- head corresponding to the velocity V with which the ' V 2 \ water leaves the nozzle I . j and the height to which the water is now capable of rising at h ~&~ ^~d* Let , = h n , be the pressure-head at the entrance to the lozzle. Then the effective head at the same point 7,2 J72 = h A -- = (i +m'). r Hence It will be observed that the delivery from the nozzle is less than that from the pipe before the nozzle was attached, but that the velocity-head at the nozzle-outlet is enormously increased. The actual height to which the water rises on leaving a nozzle is less than the calculated height, owing to NOZZLES. 177 air-resistance and to the impact of particles of water as they fall back. The force required to hold the nozzle is evidently g g 4 If the water flowing through a pipe, or hose, of length / ft. , with a velocity of v ft. per second, is quickly and uniformly shut off by a stop-valve in t sec., the pressure in the pipe near wlv the valve is increased by an amount - Ibs. per square foot. Of two forms of nozzle in general use, the one (Fig. 105) is a surface of revolution with a section which gradually diminishes to the outlet, while the other (Fig. 104) is a frustum FIG. 104. FIG. 105. of a cone, having a diaphragm with a small circular orifice at the outlet. Denoting the former by A and the latter by B r the following table gives the results of Ellis's experiments: Height of jet from i-inch Nozzle. Height of jet from ij-inch Nozzle. Height of jet from ii-inch Nozzle. Pressure in Ibs. Head in per sq. in. feet. A B A B A B IO 23 22 22 22 22 23 22 20 46 43 42 43 43 43 43 30 69 62 61 63 62 63 63 40 9 2 79 78 81 79 82 80 50 H5 94 9 2 97 94 99 95 60 138 108 104 112 1 08 H5 no 70 161 121 H5 125 121 129 123 8c 184 131 124 J 37 131 142 135 90 207 1 140 132 148 141 154 146 100 230 148 136 157 149 164 155 The coefficients of discharge for smooth cone nozzles are, very approximately, .983 for a }-in., .982 for a ^-in., - for a i -in., .976 for a ij-in. , and .971 for a ij-in. nozzle. TABLE OF FRICTION AL LOSSES IN HOSE. Freeman proposed the ij-in. nozzle shown by Fig. 106 FIG. 106. as a standard with a coefficient of discharge = .977. The coefficient of discharge for a square ring nozzle is about .74. FREEMAN'S TABLE SHOWING COMPARATIVE FRICTIONAL LOSS IN VARIOUS KINDS OF HOSE. The comparison is made on the basis of a flow of 240 gals, per min., which is about the quantity discharged by a ii-in. nozzle under a pressure of 40 Ibs. per square inch at base of play-pipe. r; U . ^3 3 2 u bo c B rt J! e/3 U) S = .52^ 1 1> 5 *o 0- . y" " Sii fc Cu . r qj flj ^ C-i o 3 w c Character of Hose. O CJ CQ C ig| i'! hij p G K C ^ 3 ^ v a O U V. y OJ T3 -J *- u 5 Q g I | ^t l^gl a|| sii ^ c 0*2"^ D ^ ^* ^ ^ *S s> o) 0-~ C u .2 > u OJ 3*0 5 C Q- U Q<5 5c S^Q S a. Q * o'" c2 ,3 s i\" solid rubber hose, extra heavy, smooth and free from ridges 2 . 52" 2 6c" / IO.O i T1 ^ nfi 25" solid rubber hose lighter than preceding and not so carefully made 2. 53 2.60 +22 aj" woven cotton hose, rubber-lined, regular 5 ' 14 .0 14 5^ heavy fire-department hose . 2 57 15 . o 6 Mr J rfi m v\" woven cotton hose, rubber-lined, lighter ' * 47 4 ' than preceding, but of about the same smoothness of interior 2.47 2 49 5 T 4-5 2 14.2 I5.8l 2\" knit cotton hose, rubber-lined. A medium- weight hose 2.50 2.68 3* JI -3 +42 16.0 13.6"; az" knit cotton hose, rubber-lined. Interior ' medium smooth 2 . 5O 2.50 4 1 16.8 o 16.8 ^O y\" knit cotton hose, rubber-lined. A regular 4a ' fire-department hose. . . . . 2j" knit cotton hose, rubber-lined. Inside 2-51 2.60 if 13-9 4-22 17.0 14.50 rather rough 2.62 _ 4-27 18.3 14. 28 aj" knit cotton hose, rubber-lined. About 3 *4 *4 1 */ same as preceding, but a little heavier .... 2.51 2.69 2 7 13-5 -1-44 T Q-4 13 55 2}" leather hose 2.80 2 l 12 . 2 +76 21 C aj" woven cotton, rubber-lined, mil! hose. ' Medium thin rubber lining. . . 2J" unlined linen hose 2 48 2.50 2-53 2.60 2^ 24.1 27.2 4- 6 +22 25-5 33- 2 15- 3' I 4-5 2" woven cotton, rubber-lined hose 2 07 2. 12 4* 33 2 -56 14 6 21. 8l si" linen hose with 2" couplings I -95 2.30 49 5 -34 32-7 18.53 WA TER-MO TOR. 1 7 9 Third. If an engine, working against a pressure of p c Ibs. -per square foot, pumps Q cu. ft. of water per second through a nozzle at the end of a hose / ft. in length, then Qp c the pumping H.P. of the engine = - . The total head at the engine end of the hose = the head corresponding to the pressure / in the hose + the head required to produce the velocity of flow v W 2g- and this head is expended in overcoming the frictional resist- ance of the hose (all other resistances are disregarded) and in producing the velocity of flow V at the outlet. Hence A = / . * = 4/?i , F! W W~^~ 2g d 2g ~*~ 2g ' and therefore / _ ' 4fl v* F 2 v^ W~' d 2g~\~ 2g 2g' rtd* TrD 2 since Q = v = - F. 4 4 The pumping H.P. SwQ 3 i i 550^ 17. Motor Driven by Water from a Pipe. Let the nozzle in the preceding article be replaced by a cylinder having its piston driven by the water from the pipe. Let u the velocity of the piston per second. i8o EXAMPLE. Let/ w = unit pressure at the end of the pipe, i.e., in the cylinder. Let d m = diameter of cylinder. Then Id \ 2 velocity of flow in pipe = fj) u. Hence i, f fft \ ^-' I fft \ i * ** ftr I 7 I 7 " ' 7~ > \ (I / 2, g Ci \ d I 2,g "W other losses of head being disregarded. Ex. A 3i-in. clean pipe, 525 ft. long, leads from a reservoir with a water surface 300 ft. above datum to a point A, 187^ ft. above datum. Find (a) the height to which the water is capable of rising at A (i) if the pipe is open to the atmosphere ; (2) if it terminates in a i-in. nozzle. What (b) force is required to hold the nozzle ? If the pipe is used to supply pressure to a water-engine with a 28-in. cylinder, determine (c) the maximum power which can be developed and the corresponding velocity of flow in the pipe. In the latter case, what (d) is the total pressure on the piston ? Take into account the resistance at the pipe entrance and assume/ = .005. Let v and V be velocities of flow in pipe and from nozzle, respect- ively. (a) i. 300 187^ = 1 12 = total effective head _ 4 x .005 x 5: 3i 12 and = 3 ft. = height to which water can rise. d 2-v = ~: and II2 . = ! + F V' \V 4 x .005 x 5 25\ _ V 2985 11 ** -I T -rriTr] I > T Therefore = ii2 x = 90.49 ft. = height to which water can rise. SIPHONS. 181 Force = momentum 62^ 22 I /_LV/^a_ I2 5 ii i ., 22$ 2401 ~32~'T ~4V~i) "64 '14 144' ' ~ ' 2985 Xt s * Let p be the pressure in pounds per sq. ft. at A. Then or Hence theH.P. = Z* = / 3 _^2_2i/ 3 _iv v 3r/7 4 Vi 2 y 4 55 550 = {IP 7~1, 3072^ 64/' which is a max. when 3 ~- = o, or v = 8 ft. per sec., and the max. H.P. = 4^! = 4.557. Also p = 62^ . 75 = 4687^ Ibs. per sq. ft., and total pres. on piston = 46874 x . .[] io T lk tons. 7 4 \i2j 2000 18. Siphons. A siphon is a bent tube, A BCD, Fig. 107, and is often employed to convey water from one reser- voir to another at a lower level. Let h^ , h 2 , respectively, be the differences of level between the top of the siphon and the entrance A and outlet D to the siphon. Then, so long as the height h^ does not exceed the head of water ______ {= 32.8 ft.) which measures FlG - ID 7- the atmospheric pressure, the water will flow along the tube in the direction of the arrow, with a velocity v given by the equation 182 INSERTED SIPHONS. I being the length of the tube A BCD, and all resistances, except that due to frictional resistance, being disregarded. If //j > 32.8 ft., each of the branches AB and DC becomes a water-barometer, and the siphon will no longer work. Even when the siphon does work, an arrangement must be made for withdrawing the air which will always collect at the upper part of the siphon. 19. Inverted Siphons. The existence of a cutting or a valley sometimes renders it necessary to convey the water from a course AB to a. course DE by means of an inverted siphon BCD of length /. Let u be the velocity of flow in AB, and /^ the height of B above a datum line. Let v be the velocity of flow in the siphon, and 7z 2 the height of D above datum. FIG. 108. Then h v h 2 = loss of head at B + frictional loss of head in siphon -f- loss of head at D ^! \^IL \ ~ 2g ' d 2g ~"~ 2g = j , approximately, assuming the entrance and outlet to the siphon formed in such iP v* a manner as to considerably reduce the losses and , and AIR IN A PIPE. 183 to allow of these losses being disregarded without practical error. Find, by chaining along the ground, the length of the siphon from B up to a point F not far from D. Call this length /j , and let // 3 be the height above datum of F, obtained with a level. Generally speaking, DF is nearly always of uniform slope. Call the slope a. Then, DF = But cosec a. = h l h^ DF . sin a, an equation from which DF can be found, as h^ /i 3 can be determined by means of a level. 20. Air in a Pipe. The effect of an air-bubble in a pipe ABCD may be discussed as follows: Let the air occupy the portion BC of a pipe. Let the surface of the water in the reservoir supplying the pipe be h^ ft. vertically above , and /z 2 ft. above D. FIG. 109. Also, let h^ be the difference of level between C and D, h the difference of level between B and C, and / the thickness of the water-layer EF. Let //"designate the head equivalent to the elastic resistance of the air in BC. Then, approximately, P 4./^i V* 1 + w ' ~d ~2g i8 4 and FLOW IN PIPE OF VARYING DIAMETER. w -= -- d 2g /! being the length of the portion of pipe from A to E, and / 2 the length from E to D. Adding the two equations, / being total length of pipe. But h^ t + ^ , very nearly. Hence an equation showing the variation of v with a variation in the height h of the space occupied by the air. NOTE. H of course varies with the temperature. 21. Flow of Water in a Pipe of Varying Diameter. The variation in the diameter is supposed to be so gradual that the fluid filaments may still be assumed to flow in sensible parallel lines. Consider a thin slice of the moving fluid, bounded by the trans- verse sections AB, CD, distant s and s + dsj respectively, from an ~ origin on the axis of the pipe. Let / be the mean intensity of pressure, A the water area, P the wetted perimeter for the section AB. Let these symbols become / + dp, A + dA, P + dP, respectively, for the section CD. Let z be the height of the C. of G. of the section AB above datum. Let z -f- dz be the height of the C. of G. of the section CD above datum. FIG. no. FLOW IN PIPE OF VARYING DIAMETER. 185 Let u y u + du be the velocities of flow across the sections , CD, respectively. Then The rate of increase of momentum of the slice ABCD in the direction of the axis momentum generated by the effective forces acting upon the slice in the same direction. iv du w The acceleration in time dt = Au . dt~ = Au . du. g dt g The total pressure on AB = / . A, and acts along the axis. The total pressure on CD (p + dp)(A + dA), and acts along the axis. The total normal pressure on the surface A CBD of the pipe (dr\ I dp \ r -\ -- )!/+ \AC 2nrp . AC, very nearly. The component of this pressure along the axis = 2 nrpA C . sin 6 = 2 npr . dr, nearly, 6 being the angle between AC and the axis. Thus the total resultant pressure along the axis pA (p -f. dp)(A -f dA) + 2npr . dr = p . dA A . dp -j- 2 npr . dr = -A.dp, since A = nr 2 , and therefore dA 2nr . dr. The component of the weight ^/"the slice along the axis = f A H -- \ds .wsmi -- ( A -\ -- \w . dz = wA . dz. The frictional resistance = P . AC . F(u) = P . ds . F(it), very nearly. Hence - = A . dp - wA . dz P . ds . F(u\ 1 86 EQUIVALENT UNIFORM MAIN. and therefore , .dp , u. du P F(u\ dz-\-~- A \- "-tds o. w g ' A w Integrating, . / C P F(u), ' H H + / ^ = a constant. 1 w 2g ' J A w Talr<= f clKC / W 2g 2g Then W 2g / gn< = a C0nstant The integration can be effected as soon as the relation between r and s is fixed. Example. Take r = a + bs y and assume f and Q to be constant. Then / , u* , i/(? Cdr \-j ^o I -r = W ig b gti J r 5 ^ + -. + T^ + r T2 / 7T : = a constant, and therefore p u* i /> 2 i ^ + ~- H h -T -^ -i = a constant. 22. Equivalent Uniform Main. A water-main usually consists of a series of lengths of different diameters. As a first approximation the smaller losses of head due to changes of section, etc., may be disregarded, and the calcula- tions may be further simplified by substituting for the several lengths a single pipe of uniform diameter giving the same fric- tional loss of head. Such a pipe is called an equivalent main. EQUIVALENT UNIFORM MAIN. 187 Let /! , / 2 , / 3 be the successive lengths of the main. FIG. in. Let d lt d ZJ d^ be the diameters of these lengths. Let v l , v 2 , v^ be the velocities of flow in these lengths. Let ^, ^ 2 , h^ be the frictional losses of head in these lengths. Let L, d, v, h be the corresponding quantities for the equivalent uniform main. Then and therefore Hence _ rf, TV ^3 ? i ^7" ^TI. 3 ~1 where it is assumed that /is the same for the several lengths of the main and also for the equivalent pipe. But nd* nd Hence an equation giving the diameter d of an equivalent pipe having the same total frictional loss of head. 1 88 BRANCH MAIN OF UNIFORM DIAMETER. Ex. What must be the diameter of a uniform pipe which may be substituted for a line of piping consisting of an 8oo-ft. length of 12-in. pipe and a 2Oo-ft. length of 6-in. pipe? 800 + 200 800 200 -y- = -F- + a/ = 720Dl <*' = 3 i and therefore d = .6738 ft., or about 8 ins. 23. Branch Main of Uniform Diameter. In a branch main AB of length L and diameter d, receiving its supply at A, Let Q w be the way-service, i.e., the amount of water given up to the service-pipes on each side. Let Q e be the end-service, i.e., the amount of water dis- charged at the end B. Then it may be assumed, and it is approximately true, that the way-service per lineal foot, viz. , =~ , is constant. Thus the amount of water consumed in way-service in a length AC of the main, where BC = s y is while the total amount of water flowing across the section of the pipe at C -L 4 v being the velocity of flow at C. Now dh, the frictional loss of head at C for an elementary length ds of the pipe, is given by the equation d 2g SPECIAL CASES OF PIPE-FLOW. 189. Integrating, the total loss of head is SPECIAL CASES. CASE I. Let Q e ' be the total discharge for the same fric- tional loss of head, h, when the whole of the way-service is stopped. Then ^O'*=h = and therefore Hence Q.' 2 > (O. + % )' and < (fi, + ^, and QJ lies between Q e -j- ~^ and Q e -| JL , its mean value being Q e +.S$Q w . CASE II. If there is no end-service, all the water having been absorbed in way-service, Q e o, and therefore Q e ' = and T fT O 2 h = CASE III. If Q t = o, fO 2 dk = 9 -,^ T 9 s*ds = elementary frictional loss of head. 52 * SPECIAL CASES OF PIPE-FLOW. Integrating between o and j, and the vertical slope, or line of free pressure, becomes a cubical parabola. CASE IV. Let the main receive its supply at A from a reservoir X in which the surface of the water is h l above datum, and let it discharge at the end B into a reservoir Y with its surface //., above datum, Fig. 114. Since (0/) a = a 2 + QeQ w + ^, therefore <2, = o; and if G w > ^3<2/ ^en the reservoir K will furnish a portion of the way-service. Suppose that X gives the supply for the distance AO (= / L ) and that Y supplies BO (= / 2 ). Let z be the height above datum of the surface in a pressure column inserted at 0. Then, neglecting the loss of head at entrance, 2 / 3 = loss of head between A and <9 = - and i fO 2 t s = loss of head between .# and 6> = - J ^ *. 3 7f*d*L* Also, /i + /, = /:. PROBLEM OF THREE RESERVOIRS. 191 24. Three Reservoirs at Different Levels connected by a Branched Pipe. Let a pipe DO of length ! { ft. and radius r l ft., leading from a reservoir A in which the water stands // t ft. above datum, divide at O into two branches, the one, OE, of length /., ft. and radius r., ft. , leading to a reservoir B in which the water stands // 2 ft. above datum, the other, OF, of length / 3 ft. and radius r s ft. , leading to a reservoir C in which the water stands h., ft. above datum. FIG. ii2. Let i\ , v 2 , ^ 3 be the velocities of flow in DO, OE, OF, respectively. Let Q^ Q 2 , Q 3 be the quantities of flow in DO, OE, OF, respectively. Let z be the height above datum to which the water will rise in a tube inserted at the junction. Two problems will be considered, and all losses of head excepting those due to frictional resistance will be disregarded. PROBLEM I. Given //,, // 2 , h^ r^, r 2 , r 3 ; to find Q lt Q 2 , > 3 ; v lt v 2 , v s , and s. Taking - = a , o * 192 PROBLEM OF THREE RESERVOIRS. For the pipe OE,^f^=c& . . (3) " Q 1 = xr 2 \. . . (4) 1 2 r 2 " " " OF, =*=*.. ( S ) .. 'Q^Krfr. . . (6) *3 *3 Also, 0,= <2 2 +<2 3 ....... (7) From these seven equations the seven required quantities can be found. In equations (3) and (7) the upper or lower signs are to be taken according as the flow is from O towards E or from E towards 0. This may be easily determined as follows : Assume z = h 2 , and then find v l and v z by means of equa- tions (i) and (5), and hence Q l and Q 3 by means of equations (2) and (6). If it is found that Q l > g s > then the flow is from to E, and equations (3) and (7) become '-="- and ft = ft'+^; while if (2i < GS > the fl w i s from to 6>, and the equations are 7 2 * -L-^lA and = . . It is assumed that f= J is the same for each pipe. <*> SPECIAL CASE. (Fig. 113.) Suppose the pipe OE closed Bt. Also, let r l = r 2 =. r 3 r, and let V be the velocity of flow from A to (7. The * l plane of charge ' ' for the reservoir A is a horizontal plane MQ distant ~ from the water-surface, / being the atmospheric pressure. V PROBLEM OF THREE RESERVOIRS. 193 The " plane of charge " for the reservoir C is a horizontal plan-e TS distant from the water- surface. w F 2 In the vertical line VTQ, take TN = and join MN. o Then, neglecting the loss of head at entrance, MN is the 4 * line of charge," or hydraulic gradient, for the pipe DF, and is approximately a straight line. Let the ' ' plane of charge ' ' KK for the reservoir B, distant from the water-surface, meet MN in G. w If the junction is vertically below 6", there is no head" FIG. 113. available for producing flow either from E towards O or from O towards E, and hydrostatic equilibrium is established. If the junction O is on the left of G, and a vertical line OKHL is drawn intersecting KK, MN, and MQ in the points K, //, and Z, there is the head HK available for producing flow from O towards E. 194 PROBLEM OF THREE RESERVOIRS. If the junction O is on the right of G, and the vertical line OHKL is drawn, the head HK is now available for producing; flow from E towards O. Let the vertical through G meet MQ in P, and take PG = Y. Then, approximately, / MG PG F QN and therefore If HL < F, the flow is from O towards E. If HL > F, " " " " " O. Again, and therefore, approximately, (O Next assume the junction O to be on the left of G, and open the valve at E. Then ( 2 > ^ = 4; (3> (4) ORIFICE FED BY TWO RESERVOIRS. 19$ and G 1 =(2 2 +G 8 , or * v l = 7' 2 '+ 7/ s . Thus 7<4+4) - *!-*= "(A", 2 + W) = 7 j 4(" 2 + "JM- W } ; and therefore ,V, + 4) + 2'w, + to' - d + 4) ^ 2 = o. Hence, assuming z- 2 to be very small as compared with F, or where Q = rrr 2 V, Thus it appears^that if a quantity Q 2 of water is drawn off" by means of a branch from a main capable of giving a total end-service Q, this end-service will be diminished by j-(2 2 , i<2., , J<2 2 , etc., according as the junction O divides the pipe DF into two portions in the ratio of I to I, I to 2, I to 3, etc. NOTE. The more correct value of i> is o and the maximum value of 71 I--.-, does not exceed -. (A -t- 4) 4 Orifice Fed by Two Reservoirs. Neglect all losses of head except the losses due to frictional resistance. When the valve at O is closed the flow is wholly from A to C y and the delivery is = ORIFICE FED BY TWO RESERVOIRS. The line of charge (hydraulic gradient) is MN, where =-^ = NV. w FIG. 114. Open the valve a little: a volume Q. z will now flow through , and a volume <2 3 into C, where a = Q - . + The " line of charge " becomes the broken line M\N. As the opening of the valve continues, the pressure-head at O diminishes, and when it is equal to // 3 -[-^ - the line of charge [$M2N\ 2N being horizontal. Hydrostatic equilibrium Is now established between O and C t and the whole of the water from A passes through O, the delivery being given by ^Opening 6^ still further, both reservoirs will serve the corifice, and the line of charge will continue to fall. EXAMPLE. When the valve is full open the ' ' line of charge ' ' is \vhere 6* = 197 , and the discharge is The- supply from A is equal to that from C when ~ -~. A ^3 The above investigation shows the advantage of a second reservoir in emergent cases when an excessive supply is suddenly demanded, as, e.g., on the occasion of a fire. Ex. A <24-in. pipe AB, 6000 ft. long, connects two reservoirs, the dif- ference of level between the water-surfaces being 250 ft. From a junc- tion O between A and B a 12-in pipe, C, 3000 ft. long, connects with an intermediate reservoir having its water-surface 150 ft. above that of the lowest reservoir. Discuss the distribution (a) when AO = 2000 ft. ; () when AO = 4000 ft.; and find (c) the position of O so that there ma- be no flow in OC, FIG. 115. FIG. 116. Take the lowest water-surface as the datum plane. Also assume that a J- .0002. r 19$ EXAMPLE. If a piezometer is inserted at O, the water will rise in it to a height 2 above datum. . Then (a) Fig. 115: Between A and O 250 z 2000 Between O and B I 50 T Z __ 7', 5 _ a 3000 ^ ~ Between O and C Z "3* = a = acvi. (I) 4000 I To find the direction of the flow in OC, let z = 150, then z/ a = o, av* ^, ar/3 9 = -\, and therefore 7/1 > 7/3. Thus more water flows from A to O than is required for the lowest reservoir, and a portion must flow to the intermediate reservoir. Hence z > 150 ft., and Therefore ~ * 20000: 4 ' 6ooo 40000: or |/ 7 i5oo 62 = \/2z 300 + Vy. By trial this gives z = 161 ft., very nearly, and then, substituting in eqs. (I), 7/1' == 222.5, or ^i = 14.916 ft. per sec., W = 9i, or v., = 3.027 " 7/ 3 a = 201.25, or 7 '3 14.186 " " Hence, also, 22 2 2 , = . x 14.916 46.879 cu. ft. per sec., 7 4 ' = ' x 3 ' 027= 2 ' 378 " 3 = . x 14.186 -44-584 " 7 4 and > + gs = 46.962 = Qi, very nearly. EXAMPLE. 199 Fig. 1 16: Between A and 250 3 -^ - = 4000 Between O and B 3000 Between (9 and C (ID 2OOO If 2 150, 7/2 = o, avi* = V and ttz/ s 2 = A- Tnus ^ > ^ and there- fore Q 3 > Qi, so that more water flows to the lowest reservoir than is supplied by the highest reservoir. Hence the balance must come from the intermediate reservoir and z < 150 ft. AISO, 0i + <2a = 03 > 7.' a + -- = 7/s. Therefore 40000: 4 0000 2000 or -1/1500 62 + 1/300 22 = }/6z. 4 By trial this gives z 96, very nearly, and then, substituting in r *> ^ 3 > *'i Vj.'^s* '* As before, let s be the pressure-head at O. Then 2 , , , '- '=tf_L . . . (i) and 0,= *^,; . . (2) /i . rj 4 ^2 (3) " <2 2 = *rfv t ; . . (4) These six equations contain the seven required quantities,, viz., fj, r 2 , f s , z/j, ^ 2 , ^ 3 , and ^. Thus a seventh equation must be obtained before their values can be found. This equation is given by the condition ' ' that the cost of the piping- laid in place should be a minimum," it being assumed that the cost of a pipe laid in place is proportional to its diameter. Hence l l r l -\- / 2 r 2 -|- / 3 r 3 = a minimum. . . . (7) From equations (i) and (2), "(6), Differentiating these three equations, dz laQ? 2 a 5> / f " " *r* ' MAINS WITH SEVERAL BRANCHES. 2O But, by equation (7), Hence which is the seventh equation required. This last equation may be written in the forms and e._ ,0.4. Q, V" t *,' + . r 25. Mains with any Required Number of Branches. Let there be n junctions and m pipes. Let ^, Aj, . . . h m be the m pressure-heads at the end of each successive length of pipe. Let z lt z 2 , . . , z n be the n pressure-heads at the 1st, 2d, 3d, . . . nth junctions. Let / n / 2 ,.../ be the lengths of the m pipes. PROBLEM I. Given k lt // 2 , . . . h m , r lt r 2 , . . . r m \ to find v lt v 2 , . . . v m1 z l , z 2 , . . . Z H . k -^ z V* There are m equations of the type -. = a . Also, the quantity flowing through the first portion of the main is equal to the sum of the quantities flowing through all the branches at the first junction, and an analogous equation will hold for each of the remaining n I junctions. Thus n additional equations are obtained. From these m -J- n equations 7^ 7> 2 , . . . v m , s l , z 2 , . . . z n may be found analytically or by the method of repeated approximation. 202 VARIATION OF VELOCITY IN. TRANSVERSE SECTION. PROBLEM II. Given /i lt /i 2 , . . . //,, Q l , Q 2 , . . . Q m \ to find r lt r 2 , . . . r m , ^ , z^ . . . z n . There are now only m equations of the type h / = a r involving m -(- n unknown quantities, and the problem admits of an infinite number of solutions. It is therefore assumed that the cost of the piping laid in place is to be a minimum. Thus n new equations are obtained, and the m -\- n equations may be solved analytically or by repeated trial. NOTE. The maximum velocity of flow in town mains is from 2 to 7 ft. per second. 26. Variation of Velocity in a Transverse Section. Assumption. That the water in any portion of a pipe is made up of an infinite number of hollow concentric cylinders of fluid, each moving parallel to the axis with a certain definite velocity. Let u be the velocity of one of these cylinders of radius x and thickness dx. Then the flow across a transverse section FIG. 117. *~ is given by the equation dq = 27tx dx . u y and the total flow (i) r being the radius of the pipe. If v m be the mean velocity for the whole transverse section of the pipe, Q (2) VARIATION OF VELOCITY IN TRANSVERSE SECTION. 203 Again, assuming with Navier that the surface resistance Between two concentric cylinders is of the nature of a viscous resistance and may be represented by k per unit of area at the radius x, k being a coefficient called the coefficient of viscosity, then the total resistance at the radius x for a length ds of the cylinder .du du = 2n x . ds . k-j- = 2nk . ds . x-j-. dx dx The total resistance at the radius x + dx r du d( du\ ,-\ _j_ 2nk . ds\ x -j- -j-\x-r }dx . L dx* dx\ dxl J Hence the total resultant resistance for the length ds of the cylinder under consideration d I du\ = 2nkds- r \x- r \dx. dx\ dx} . The component of the weight of the slice of the cylinder in the direction of the axis = w . 2 nx . dx . ds . sin 0, 6 being the inclination of the axis to the horizon. Let dz be the fall of level in the distance ds. Then ds = ds . sin B. Therefore component of weight in direction of axis = w . 2 xx dx . ds. 204 VARIATION OF VELOCITY IN TRANSVERSE SECTION. The resultant pressure on the slice in the direction of motion = \P (f + dp)\2nx . dx = 2nx . dx . dp. Then, since the motion is uniform, w . 2 nk . ds . ~T~(^~j-)dx w .2nx . dx . dz2nx . dx . dp O, and therefore k . ds d I du\ dp -j- 1 * j 1 dz -- - = o. x dx\ dx) w Integrating only for the cylinder under consideration, ks d I du\ I p\ ~~j~\ x ~r ) \s -4- = a constant. x dx\ dxl \ v w) But z -f- is evidently independent of x and is a linear function of s (Art. 5, Chap. II). Hence i dL du\ T\XT~\ = a constant = A* suppose. x dx\ dxl Therefore Integrating, dx Assuming that the central fluid filament is the filament of maximum velocity, then when x = o, -j- is also nil. Therefore du Ax* B = o, and x-j- = - , dx 2 VARIATION OF VELOCITY IN TRANSVERSE SECTION. 205 and therefore du * Z= : A 2 -.. (4) Integrating, Eq. 4, C being a constant of integration. Since dp is the difference of intensity of pressure on the ends of the cylindrical slice, du 2nx . ds . k-r = nx* . dp 7tx*w . dh. Therefore du ivx dh wxi dx ~ 2k ds 2k 1 and, by equation (4), wi Let max be the velocity of the central filament, i.e., the value of u when x = o. Then and ..-*= --x* = Dx\ . . . . (5) where D = 4 Again, by equation (i), .dx 206 VARIATION OF VELOCITY IN TRANSVERSE SECTION. and by equation (2), Dr* ( 6 ) If w, = velocity at pipe wall, then, by equation (5), * = ^inax. - /V ...... (7) Hence, by equations (6) and (7), U 5 + fc m ax. = *V M . ..... (8) If u = o when x r y then C = A , and 4 K- Therefore Anr* winr* NOTE. In a paper by Gardner S. Williams and others, in the Pro- ceedings of the Am. Soc. of C. E. for May, 1901, giving the results of experiments on the flow of water in pipes, the inferences are made . that at ordinary velocities of flow, and under normal conditions, the ratio of the mean velocity to the maximum 13.84; that in a straight pipe there will be, under some conditions, a difference of pressuie at different points in the circumference of the same cross-section ; that the normal curve of velocities is an ellipse; that the effect of a flow disturbance is felt many diameters beyond the point at which it occurs ; that for a maximum flow careful alignment is as necessary as a smooth interior. WATER-METERS. 207 27. Gauging of Pipe-flow. A variety of meters have been designed to register the quantity of water delivered by a pipe. The principal requisites of such a meter are : 1. That it should register with accuracy the quantity of water delivered under different pressures. 2. That it should not appreciably diminish the effective pressure of the water. 3. That it should be compact and adaptable to every situation. 4. That it should be simple and durable. The Venturi Meter (Fig. 118) is so called from Venturi, who first pointed out the relation between the pressures and velocities of flow in converging and diverging tubes. FIG. 118. As shown by the longitudinal section, Fig. 1 19, this meter consists of two truncated cones joined at the smallest sections by a short throat-piece. At A and B there are air-chambers with holes for the insertion of piezometers, by which the fluid pressure may be measured. By Art. 5, Chap. I, the theoretical -quantity Q of flow through the throat at A is a,a. y so 8 WATER-METERS. # ! , a 2 being the sectional areas at A and B, respectively, and H 2 HI the difference of head in the piezometers, or the "head on Venturi, " as it is called. FIG. rig. Introducing a coefficient of discharge C, the actual delivery through A is Q = C An elaborate series of experiments by Herschel gave C values varying between .94 and 1.04, but the great majority of the values lay between .96 and .99. FIG. 120. Schonheyder's Positive Meter. FIG. 121. The Universal Meter. The piezometers may be connected with a recorder, and thus a continuous register of the quantity of water passing through the meter may be obtained at any convenient position within a radius of 1000 ft. This distance may be extended to several miles by means of an electric device. WATER-METERS. 209 Other meters may be generally classified as Piston or Reciprocating Meters and Inferential or Rotary Meters. They are all provided with recorders which register the delivery with a greater or less degree of accuracy. The piston meter (Fig. 120) is the most accurate and gives a positive measurement of the actual delivery of water as recorded by the strokes of the piston in a cylinder which is filled from each end alternately. Thus an additional advantage FIG. 122. The Buffalo Meter. FIG. 123. The Union Rotary Piston Meter. possessed by a water-engine is that the working cylinder will also serve as a meter. In inferential meters a drum or turbine is actuated by the force of the current passing through the pipe, but it often happens that when the flow is small the force is insufficient to cause the turbine to revolve, and there is consequently no register of the corresponding quantity of water passing through the meter. 210 EXAMPLES. EXAMPLES. (N.B. Take^ 1 = 32 and 6 gallons = i cu. ft. unless otherwise specified.) 1. A water-main is to be laid with a virtual slope of i in 850, and is to give a maximum discharge of 55 cubic feet per second. Determine the requisite diameterof pipe and the maximum velocity, taking/=.oo64. Ans. 3.679 ft.; 3.2888 ft. per sec. 2. Find the loss of head due to friction in a pipe ; diameter of pipe = 12 in., length of pipe = 5280 ft., velocity of flow = 3 ft. per second ; f = .0064. Also find the discharge. Ans. 19.008 ft.; 2.3562 cu. ft. per sec. 3. A pipe has a fall of 10 ft. per mile ; it is 10 miles long and 4 ft. in diameter. Find the discharge, assuming/ = .0064. Ans. 54.7 cu. ft. per sec. 4. A pipe discharges 250 gallons per minute, and the head lost in friction is 3 ft. Find approximately the head lost when the discharge is 300 gallons per minute ; also find the work consumed by friction in both cases. Ans. 4.32 ft.; 7500 ft.-lbs.; 12,960 ft.-lbs. 5. What is the mean hydraulic depth in a circular pipe when the diameter . water rises to the height above the centre ? 2 V 2 10 Ans, x diameter. 33 6. A 12-inch pipe has a slope of 12 feet per mile ; find the discharge. (/ = .005.) Ans. 2.118 cu. ft. per. sec. 7. The mean velocity of flow in a 24*in. pipe is 5 ft. per second; find its virtual slope, f being .0064. Ans. i in 200. 8. Calculate the discharge per minute from a 24-5n. pipe of 4000 ft. length under a head of 80 ft., using a coefficient suitable for a clean iron pipe. Ans. 34.909 cu. ft. per sec. 9. How long does it take to empty a dock whose depth is 31 ft. 6 ins. and which has a horizontal sectional area of 550,000 sq. ft., through two 7-ft. circular pipes 50 ft. long, taking into account resistance at en- trance ? Ans. 214 min. 6 sec. 10. The virtual slope of a pipe is i in 700; the delivery is 180 cubic feet per minute. Find the diameter and velocity of flow. Ans. i. 26 ft.; 2.401 ft. per sec. 11. Determine the diameter of a clean iron pipe, 100 feet in length, which is to deliver .5 cu. ft. of water per second under a head of 5 feet. Assume/ = .006. Ans. .328 ft. 12. A reservoir of 10,000 sq. ft. area and 100 ft. deep discharges t EXAMPLES. 211 through a pipe 24 ins. in diam. and 2000 ft. in length. Find the velocity of flow. What should the diam. be in order that the reservoir may be emptied in two hours? (/=.oo64.) Ans. 15.37 ft. per sec.; 4.0923 ft. 13. The pressure from an accumulator at the entrance of a df-in. pipe L ft. long is looo Ibs. per sq. in. If A^ is the total H.P. available at the N \*L inlet, show that the H.P. absorbed in frictional resistance isf /being - = .0081. IS 14. The delivery at the end of a 3-inch pipe is 11.06 H.P. The total effective head at the entrance to pipe is 896 feet. The loss in frictional resistance is 21 per cent. Find the distance to which the energy is transmitted. Ans. 15,000 ft.,/ being .0064. ^15. A reservoir has a superficial area of 12,000 ft. and a depth of 60 ft.; it is emptied in 60 minutes throuali/0r horizontal circular pipes, equal in diameter and 50 ft. long. Fiftd the diameter. (/ = .0064.) Ans. 1.786 ft. Explain how the total head is made up, and draw the plane of charge. 16. A 3-inch pipe is very gradually reduced to ^ inch. If the pres- sure-head in the pipe is 40 ft., find the greatest velocity with which the water can flow through. Ans. 1.4 ft. per sec. 17. Water flows through a 24-inch pipe 5000 yards in'length. At 1000 yards it yields up 300 cubic feet per minute to a branch. At 2800 yards it yields up 400 cubic feet per minute to a second branch. At 4000 yards it yields up 600 cubic feet per minute to a third branch. The de- livery at the end is 500 cubic feet per minute. Find the head absorbed by friction. (/ = .0075.) Ans. 177.801 ft. /^ 18. Find the H.P. required to raise 550 gallons per minute to a height of 60 feet, through a pipe 100 feet in length and 6 in. in diameter, the coefficient of friction being .0064. Ans. 10.74. 19. What head of water is required for a 5-in. pipe, 150 ft. in length, to carry off 25 cu. ft. of water per minute ? Ans. 1.56223 ft. What head will be required if the pipe contains two rectangular knees? Ans. 1.84918 ft. 20. Determine the delivery of a 2-in. pipe, 48 ft. long, under a 5-ft. head, /being .005. Ans. .1449 cu. ft. per sec. What will be the delivery if the pipe has 5 small curves of 90 cur- vature, the ratio of the radius of the pipe to that of the curves being 1:2? Ans. .1381 cu. ft. per sec. 21. The curved buckets of a turbine form channels 12 in. long, 2 in. wide, and 2 in. deep ; the mean radius of curvature of the axis is 8 in.; the water flows along the channel with a velocity of 50 ft. per minute. What is the head lost through curvature? Ans. .00138 ft. 22. Find the power transmitted by water flowing at 80 ft. per sec. in a 36-inch pipe, the metal being i| inches thick and the allowable stress 212 EXAMPLES. 2800 Ibs. per square inch. If the pipe is i miles in length, find the loss of power. Ans. 576 H.P.; 720.2 ft.-lbs. 23. Find the diameter of a pipe \ mile long to deliver 1500 gallons of water per minute with a loss of 20 feet of head, (f = .005.) Ans. 1.0135 ft- 24. Water is to be raised 20 ft. through a 3o-ft. pipe of 6 in. diameter. Find the velocity of flow, assuming that 10 per cent of additional power is required to overcome friction, and that/ = .0075. Ans. 8.44 ft. per sec. 25. In a pipe 3280 ft. in length and delivering 6750 gallons per min., the loss of head in friction is 83 ft. Taking/ .0064, find the diameter. Ans. 1.527 ft. 26. Calculate, by Thrupp's formula, the flow through a 4-in. rough wrought-iron pipe having a fall of 33 feet per mile. Ans. .1426 cu. ft. per sec. 27. A clean 6-in. pipe has a' virtual slope of i per 400. Taking / = .005, find the velocity of steady flow, the discharge, and the energy absorbed in frictional resistance in 1000 feet. Ans. 2 ft. per sec.; cu. ft. per sec.; 6i T \ 3 ^ ft.-lbs. 28. A 6-in. pipe, 500 ft. long, discharges into a 3-in. pipe, also 500 ft. long. The effective head between the inlet and outlet is 10 feet. Find the discharge, taking / = .0064, and making allowance for the resistance at the inlet. Ans. .1703 cu ft. per sec. 29. How far can 100 H.P. be transmitted by a 3|-in. pipe with a loss of head not exceeding 25 per cent under an effective head of 750 Ibs. per square inch ? Ans. 5426.3 ft. 30. A pipe 2000 ft. long and 2 ft. in diameter discharges at the rate of 16 ft. per second. Find the increase in the discharge if for the last looo ft. a second pipe of same size be laid by the side of the first and connected with it so that the water may flow equally well along either pipe. Ans. 7.24 cu. ft per sec. 31. A pipe of length /and radius r gives a discharge Q. How will the discharge be affected (i) by doubling the radius for the whole length ; (2) by doubling the radius for half the length ; (3) by dividing it into three sections of equal length, of which the radii are r, , and , 2 4 respectively? (/=. coefficient of friction.) Ans. i. New discharge = 4 6 4 / 6 4 /7U 33 //; ' 3 . =, Q ( _ 9r + I2// \* ^524.7 1 2r + 4228/7; ' 32. A 24-in. pipe 2000 ft. long gives a discharge of Q cubic feet of water per minute. Determine the change in Q by the substitution for the foregoing of either of the following systems : (i) two lengths, each EXAMPLES. 213 of looo ft., whose diameters are 24 ins. and 48 ins. respectively ; (2) four lengths, each of 500 ft., whose diameters are 24 ins., 18 ins., 16 ins., and 24*ins. Draw the " plane of charge " in each case. Ans. (i) Discharge is increased 33.2 per cent taking loss at change of section into account; Discharge is increased 35.7 per cent disregarding loss at change of section. (2) Discharge is diminished 45 per cent disregarding losses at change of section. 33. Q is the discharge from a pipe of length /and radius r\ examine the effect upon Q of increasing r to nr for a length ml oi the pipe. f 3.2/7 I* 2 f+ (I -m+-\ + Ans. New discharge = Q , ^ _ ( ^ a _ r I 1 ."* 1 " 1 "**; ' 34. A 5-in. pipe, 300 ft. long, discharges into a 3-in. pipe, 200 ft. long, the total fall being 5 feet. Find the quantity of flow in gallons per hour. Ans. 4080. 35. A main, 1000 ft. long and with a fall of 5 ft., discharges into two branches, the one 750 ft. long with a fall of 3 ft., ihe other 250 ft long with a fall of i ft. The longer branch passes twice as much water as the other and the total delivery is 47} cu. ft per minute. The velocity of flow in the main is 2| ft. per second Find the diameters of the main and branches. (/ = .0064.) Ans. 63245 ft. ; .51 ft. ; .36 ft. 36. The water in a 12-in. main. 800 ft. long, flows at the rate of i ft. per second and one third of the water ; s discharged into a branch 200 ft. long with a fall of i in 40, while the remainder passes into a 6oo-ft. branch with a fall of i in 60. The effective head between the inlet and outlet of the main is i\ ft. Find the total discharge and the diameters of the branches, taking f .0064, and making allowance for loss at inlet but disregarding losses at the Junction. Ans. 94! cu. ft. per sec. , .27 ft. ; .39 ft. 37. If a pipe whose diameter is 8 ins. suddenly enlarges to one whose diameter is 12 ins., find the power required to force 1000 gallons per minute through the enlargement, and draw to scale the plane of charge. Ans. Energy expended = .1377 H.P. 38. 1000 gallons per minute are forced through a system of pipes AB, BC, CD. of which the lengths are 100 ft., 50 ft., and 120 ft., and the radii 6 ins., 3 ins., and 4 ins., respectively. Draw to scale the plane of charge. Ans. Loss in friction from A to B 14.744 ft.; ioss at B = 14.56 ft. " " " " B to C- 235.9 " : 4< " C 8.819" ' " " " C toD= 134.36 " 214 EXAMPLES. 39. A pipe 4 ins. in diameter suddenly contracts to one 3 ins. iir diameter ; find the power necessary to force 250 gallons per minute through the sudden contraction. Ans. 1.23997 H.P. 40. Water flows from a 3-in. pipe through a i^-in. orifice in a dia- phragm into a 2-in. pipe. What head is required if the delivery is to be 8 cu. ft. of water per minute? Ans. 2.826 ft. 41. 500 gallons of water per minute are forced through a continuous line of pipes AB, BC t CD, of which the radii are 3 ins., 4 ins., 2 ins., and the lengths 100 ft., 150 ft., and 80 ft., respectively. Find the total loss of head (a) due to the sudden changes of form at B and C, () due to friction. Find (c) the diameter of an equivalent uniform pipe of the same total length. Ans. (a) .1378 ft.; 1.152 ft. (ff) 3.688 ft. in AB\ 1.313 ft. in BC\ 22.393 ft - m c ^- (c) .4212 ft. 42. AB, BC, CD is a system of three pipes of which the lengths are 1000 ft., 50 ft., and 800 ft., and the diameters 24 ins., 12 ins., and 24 ins., respectively; the water flows from CD through a i-in. orifice in a thin diaphragm, and the velocity of flow in AB is 2 ft. per second. Draw the plane of charge and find the mechanical effect of the efflux, /being .0064. Ans. Loss at C -^ ft. ; at B = -jfa ft.; in friction from A to B .8 ft. ; from B to C 1.28 ft. ; from C to D = .64 ft. ; energy of jet = 14,81 if H.P. 43. 1000 gallons per minute flows through a sudden contraction from 12 ins. to 8 ins. at A, then through a sudden enlargement from 8 ins. to 12 ins. at B, the intermediate pipe AB being 100 ft. long. Draw the plane of charge, /being .0064. Ans. Loss at A = .288 ft. ; at B .281 ft. ; in friction from A to B = 3.499 ft. 44. Water flows from one tube into another of twice the diameter ; the velocity in the latter is 10 ft. Find the head corresponding to the resistance. Ans. 14.0625 ft. 45. A 2-in. pipe A suddenly enlarges to a 3-in. pipe B, the quantity of water flowing through being 100 gallons per minute. Find the loss of head and the difference of pressure in the pipes (i) when the flow is from A to B ; (2) when the flow is from B to A, Cc being .66. Ans. (i) Loss of head = 8.639 m - Gain of pressure-head = 13.83 " (2) Loss of head = 7.428 " Diminution of pressure-head = 29.88 " 46. A 3-in. horizontal pipe rapidly contracts to a i-in. mouthpiece, whence the water emerges into the air, the discharge being 660 Ibs. per minute. Find the pressure in the 3-in. main. EXAMPLES. 2 1 5 If the 3-in. pipe is 200 ft. in length and receives water from an open ta'ak, find the height of the tank,/ being .005. Ans. 1003.5 Ibs. per sq. ft. ; 19.92 ft. 47. A horizontal pipe 4 ins. in diameter suddenly enlarges to a diameter of 6 ins. ; find the force required to cause a flow of 300 gallons of water per minute through the sudden enlargement. Ans. .06 H.P. 48. 1000 gallons per minute is to be forced through a system of pipes AB, BC, CD, of which the lengths are 100 ft., 50 ft., 120 ft., and the radii 4 ins., 6 ins., and 3 ins., respectively. What must be diameter of equivalent uniform pipe ? Draw the plane of charge, /being .0064. Ans. Diameter = 3.4 ins. ; loss in friction from A to B = 1 1 1.96 ft.; loss at B = 4.499 ft.; " " " " B to C = 7.372 " " " C= 14.56 " " " " " CtQjD= 566.17 " 49. Find the H.P. required to pump 1,000,000 gallons of water per day of 24 hours to a height of 300 ft. through a line of straight piping 3000 ft. long, the diameter of the pipe being 8 ins. for the first loco ft., 6 ins. for the second, and 4 ins. for the third, allowance being made for the loss at inlet and the losses at abrupt changes of section ; also 4 is to be taken as the coefficient of resistance for pump-valves. (At changes of section c e = .64.) What is the diameter of an equivalent uniform pipe? (/= .0064.) Ans. 196; diam. = .403 ft., or say 5 ins. 50. In a given length / of a circular pipe whose inner radius is r and thickness e, a column of water flowing with a velocity v is suddenly checked by the shutting off of cocks, etc. Show that in which // = head due to the velocity v, E = coefficient of elasticity, E\ = coefficient of compressibility of water, A = extension of pipe cir- cumference due to E. 51. The water surface in one reservoir is 500 ft. above datum, and is 100 ft. above the surface of the water in a second reservoir 20,000 ft. away, and connected with the first by an i8-in. main. Find the delivery per second, taking into account the loss of head at the entrance. Ans. 7.64 cu. ft. per sec., /being .0064. 52. Determine the discharge from a pipe of 12 in. radius and 3280 ft. in length which connects two reservoirs having a difference of level of 128 ft. Take into account resistance at entrance. Draw the plane of charge. (/ = .005.) Ans. 48.571 cu. ft. per sec. 53. Determine the diameter of a clean iron pipe 5000 ft. in length which connects two reservoirs having a total head of 40 ft. and dis- charges into the lower at the rate of 20 cu. ft. per second. Draw to scale the line <.f charge. (/= .005.) Ans. 1.9219 ft. 216 EXAMPLES, 54. The difference of level between the two reservoirs is 100 ft., and they are connected by a pipe 10,000 ft. long. Find the diameter of the pipe so as to give a discharge of 2000 cubic feet per minute (a) by Darcy's formula, (b} assuming/ = .0064. (Allow for loss of head at entrance.) Ans. (a) 2.256 ft. if a= .0001622 ; (b) 2.360 ft. 55. Two reservoirs are connected by a 12-inch pipe ij miles long. For the first 500 yards it has a slope of i in 30, for the next half mile a slope of i in 100, and for the remainder of its length it is level. The head of water over the inlet is 55 ft. and that over the outlet is 15 ft. Determine the discharge in gallons per minute. (Take/ = .0064.) Ans. 1950.66. 56. Two reservoirs are connected by a 6-in. pipe in three sections, each section being three quarters of a mile in length. The head over the inlet is 20 ft., that over the outlet 9 ft. The virtual slope of the first section is i in 50, of the second i in 100, and the third section is level. Find the velocity of flow, and the delivery,/ being .005. Ans. 4.5 ft. per sec.; 332 gallons per minute. 57. A pipe 5 miles long, of uniform diameter equal to 12 in., conveys water from a reservoir in which the water stands at a height of 300 tt. above Trinity high-water mark, to a reservoir in which the water stands at a height of 150 ft. above the same datum. To what height will water rise in a supply-pipe taken one mile from the lower end ? For what pressure would you design the main at this point, if it lies 20 ft. above the level of the lower reservoir? (/ = .0064.) Ans. 179.755 ft.; 19.13 Ibs. per sq. in. 58. A clean 6-in. pipe, 1000 ft. long, has four sharp knees, viz., one of 60, two of 90, and one of 120. Find the head wasted at the knees and in the straight pipe, the flow being at the rate of 150 gallons per minute. Ans. .2734 ft.; 3.0237 ft. 59. A 6-in. pipe, 4000 feet in length", leads from a reservoir A to a. point O, at which it divides into two 6-inch branches, each 4000 feet in length, the one leading to a reservoir B, the other to a reservoir C. The surface of the water in A is 100 feet above that in B and 200 feet above that in C. Find the velocities of flow in the three branches, /being .0064. Ans. v\ = 7.89 ft. per second z/ 3 ; v^ o. 60. A pipe 24 in. in diameter and 2000 ft. long leads from a reservoir in which the level of the water is 400 ft. above datum to a point B, at which it divides into two branches, viz., a i2-in. pipe JSC\ 1000 ft. long, leading to a reservoir in which the surface of the water is 250 feet above datum, and a branch BD, 1500 ft. long, leading to a reservoir in which the surface of the water is 50 ft. above datum. Determine the diameter of BD when the free surface-level at B is (a) 300 ft.; (b} 250 ft., and (c) 200 above datum. Ans. (a) 1.454 ft.; (b) 1.783 ft.; (c) 2.096 ft. 61. Two reservoirs A and B are connected by a line of piping MON, 2000 ft. in length. From the middle point O of this pipe a branch OP, 1000 ft. in length, leads to a reservoir C. The reservoirs A and B are EXAMPLES. 2 1 7 200 feet and 100 feet, respectively, above the level of C. The deliveries in MO, OP, ON, in cubic feet per second, are V^.V-*. and * respectively. Ffnd (a) the velocities of flow in MO, OP, ON\ (b) the radii of these lengths; (c) the height of the free surface-level at O above C,/being .0064. Ans. (a) 11.121 ft. per sec. in MO ; 10.158 ft. per sec. in OP; 14.145 ft. per sec. in ON. (b) .5 ft.; .41831 ft.; .26588 ft. (c) 150.5 ft., very nearly. 62. Find the amount of water in gallons per day which will be de- livered by a 24-inch cast-iron pipe, 15,000 ft. in total length, when the water surface at the outlet is 87^ ft. below the water surface at the inlet, taking/ = .001 and allowing for resistance at inlet. If the water, instead of flowing into a reservoir, is made to drive a reaction turbine, what must be the velocity of flow in the pipe to give a max. speed? What will be the H.P. of the turbine if its efficiency is .84? A third reservoir is connected with the system by means of a 24-111. cast-iron pipe, 7500 ft. long, joined to the main at the middle point. The water surface of this intermediate reservoir is 50 ft. above that of the lowest reservoir. Discuss the distribution. Ans. 22,628, 57if; 7.7 ft. per sec. ; 12.63 H.P. ; 2 = 73.68 or 51.32 ft.; v\ 7.76 or 12.42 ft. per sec.; z/ a = 10.05 or 2 -373 ft. per sec.; 7/1 = 17.73 or ! 4-S ft. per sec. 63. The water-levels in two reservoirs A and B are, respectively, 300 ft. and 200 ft. above that in C. The reservoir A supplies 3 cu. ft. of water, of which 2 cu. ft. go to B and i cu. ft. goes to C. A pipe 2500 ft. long leads from A to a junction at O, from which two branches, each 2500 ft. in length, lead, the one to B and the other to C. Assuming that the cost of laying a pipe in place is proportional to the diam. and that this cost is to be a minimum, find the pressure head at O and the diams. of the pipes. Ans. 164 ft.; diam. of AO = .66 ft., of OB 63 ft., of OC = .4 ft. 64. An engine pumps a volume of Q cubic feet of water per second through a hose i ft. in length, and d feet in diameter, having at the end a nozzle D feet in diameter. Find the pumping H.P. and apply your result to the determination of the H.P. of an engine which is to pump 30 cu. ft. of water per minute through a i-in. nozzle at the end of a 3-in. hose 400 ft. in length (/= .00625). Also find the force required to hold the nozzle. Ans. ii 3 ^ H.P. ; 89$! Ibs. 65. A fire-engine pumps water through a 4oo-ft. length of 2^-in. bore at the rate of 12 ft. per second, and discharges through a i-in. nozzle. Find the pressure in the hose, and the pumping H.P. Also find the force required to hold the nozzle, (f == .00125.) Ans. .6702 Ibs. per ft.; 5.0916; 59.95 Ibs. 66. The conduit-pipe for a fountain is 250 ft. long and 2 in. in diam- eter ; the coefficient of resistance for the mouthpiece is .32 ; the entrance orifice is sufficiently rounded, and the bends have sufficiently long radii 2 1 8 EXAMPLES. of curvature to allow of the corresponding coefficient of resistance being- disregarded. How high will a ^-in. jet rise under a head of 30 ft. ? Ans. 20.4 ft. 67. Water surface of a reservoir is 300 ft. above datum, and a 4-in. pipe 600 ft. long leads from reservoir to a point 200 ft. above datum. Find the height .to which the water would rise (a) if end of pipe is open to atmosphere, (b) if it terminates in a i-in. nozzle. In latter case find longitudinal force on nozzle. Ans. (a) 2f ft. ; (ff) 87.52 ft.; 59.693 Ibs. 68. The surface of the water in a tank is 388 ft. above datum and is connected by a 4-in. pipe 200 ft. long with a turbine 146 ft. above datum. Determine the velocity of the water in the pipe at which the power obtained from the turbine will be a maximum. Assuming the efficiency of the turbine to be 85 per cent, determine the power, f being .005. Ans. 19.928 ft. per sec. : 27.11075 H.P. 69. A pipe 12 ins. in diameter and 900 ft. long is used as an inverted siphon to cross a valley. Water is lead to it and away from it by an aqueduct of rectangular section 3 ft. broad and running full to a depth of 2 ft. with an inclination of i in 1000. What should be the difference of level between the end of one aqueduct and the beginning of the other, /being .0064 for the pipe, and .008 for the aqueduct ? Ans. 14.39. 70. Water flows through a pipe 20 ft. long with a velocity of 10 ft. per second. If the flow is stopped in T V second and if retardation during the stoppage is uniform, find the increase in the pressure produced. (g = 32 and the density of the water = 62.5 Ibs. per cu. ft.) Ans. 62^ cu. ft. of water. 71. An hydraulic motor is driven by means of an accumulator giving 750 Ibs. per square inch. The supply-pipe is 900 ft. long and 4 ins. in diameter. Find the maximum power attainable, and velocity in pipe. (/ = .0075.) Ans^ 242.4 H.P. ; 21.203 ft - per sec. 72. A 2-in. hose conveys 2 gallons of water per second. Find the longitudinal tension in the hose. Ans. 9.18 Ibs. 73. Find the pumping H.P. to deliver i cu. ft. of water per second through a i-in. nozzle at end of a 3-in. hose 200 ft. long,/ being .016. Ans. 97.335 H.P. 74. The surface of the water in a tank is 286 ft. above datum. The tank is connected by a 4-in. pipe 500 ft. long with a 36-in. cylinder 170 ft. above datum. Find (a) the velocity of flow in the pipe for which the available power will be a maximum; () the power. If the piston moves at the rate of i ft. per minute, find (c) the pressure on the piston. Also find the height to which the water would rise if (d) the cylinder end of the pipe were open to the atmosphere and if (e) the pipe termi- nated in a nozzle i in. in diameter, neglecting the frictional resistance of the nozzle. Finally, find (/) the power required to hold the nozzle. (Coeff. of friction = .005.) Ans. (a) 8.93 ft. per sec.; () 6.85 H.P. ; (c) 22.8 tons per sq. ft. ; (d) 3.74 ft. ; (e) 103.8 ft. ; (/) 70.8 Ibs. EXAMPLES. 219 75. A 3-in. hose, 400 ft. in length, terminates in a f-in. nozzle; Water enters the hose under ahead of 297!- ft. Find the velocity of efflux, the height to which the issuing jet will rise, the pressure-head at the nozzle inlet, and the force required to hold the hose, f being .00625. Ans. 128 ft. per sec. ; 256 ft. ; 18,437^ Ibs. per sq. ft. ; 98! Ibs. 76. A reducer, 10 ft. long, conveys 400 gallons of water per min- ute, and its diameter diminishes from 12 ins. to 6 ins.; find the total loss of head due to friction. Ans. .05529. 77. A reservoir is to be supplied with water at the rate of 11,000 gallons per minute, through a vertical pipe 30 ft. high; find the mini- mum diameter of pipe consistent with economy. Cost of pipe per foot = d, d being the diameter ; cost of pumping = I cent per H.P. per hour ; original cost of engine per H.P. = $100.00; add 10 per cent for depreciation. Engine works 12 hours per day for 300 days in the year, /being .0064. Ans. 4.375 ft. 78. A city is supplied with water by means of an aqueduct of rect- angular section, 24 ft. wide, running 4 ft. deep, and sloping i in 2400. One-fourth of the supply is pumped into a reservoir through a pipe 3000 ft. long, rising 25 ft. in the first 1500 ft., and 75 ft. in the second 1500 ft. The pumping is effected by an engine burning 2| Ibs. of coal per H.P. per hour, and working constantly through the year. A percentage is to be allowed for repairs and maintenance ; the cost of the coal per ton of 2000 Ibs. is $4; the prime cost of the engine is $100 per H.P.; the efficiency of the engine is f ; the coefficient of pipe friction is .0064, the cost of the piping is $30 per ton. Determine the most economical diameter of pipe, and the H.P. of the engine, /being .0064 for the pipe and .08 for the channel. Ans. 4.84 ft.; 456.455 H.P. 79. A vessel with 500 sq. ft. of surface experiences a resistance of 150 Ibs. per sq. ft. when steaming at 5 knots. How much H.P. will be absorbed in frictional resistance by a vessel with 10,000 sq. ft. of surface steaming at 18 knots? Ans. 2140^. 80. The performances of two similarly designed ships are to be com- pared. The one, with a length of 300 ft. and a displacement of 8000 tons, is to steam at 20 knots. What should be the length and displace- ment of the other, which is to steam at 21 knots? Compare also the I.H.P.s. Ans. 33of ft.; 10,720 tons; 1.34. 81. From a central junction four mains, each 10,000 ft. long, lead to four reservoirs, A, B, C, D, the water-levels in A, B, C being 600, 400, and 200 ft., respectively, above that in D. If the diameter of each main is 12 ins., find (a) the effective head at the junction and the velocities of flow. If the velocity in each main is 5 ft. per sec., find (b) the effective head at the junction and the diameters of the mains. Ans. (a) 300 ft,; 8.66 */ s in highest and lowest mains; 5 f / s in intermediate mains. (b) 300 ft.; 4 ins. for highest and lowest mains ; 12 ins. for intermediate mains. CHAPTER III. FLOW OF WATER IN OPEN CHANNELS. I. Channel-flow Assumptions. A transverse section of the water flowing in an open channel may be supposed to consist of an infinite number of elementary areas representing the sectional areas of fluid filaments or stream-lines. The velocities of these stream-lines are very different at different points of the same transverse section, and the distribution of the pressure is also of a complicated character. Generally speaking, the side and bed of a channel exert the greatest retarding influence on the flow, and therefore along these sur- faces are to be found the stream-lines of minimum velocity. The stream-lines of maximum velocity are those farthest removed from retarding influences. If the stream-line veloci- ties for any given section are plotted, a series of equal velocity- curves may be obtained. In a channel of symmetrical section FIG. 124. the depth of the stream-line of maximum velocity below the water-surface is less than one fourth of the depth of the water, while the mean velocity-curve cuts the central vertical line at 220 CHANNEL-FLOW ASSUMPTIONS. 221 a point below the surface about three fourths of the depth of tfre water. In the ordinary theory of flow in open channels the varia- tion of velocity from point to point in a transverse section is disregarded, and it is assumed that all the stream-lines are sensibly parallel and move normally to the section with a common velocity equal to the mean velocity of the stream. With this assumption, it also necessarily follows that the dis- tribution of pressure over the section is in accordance with the hydrostatic law. Again, it is assumed that the laws of fluid friction already enunciated are applicable to the flow of water in open channels. Thus the resistance to flow is proportional to some function of the velocity (F(v)\ to the area (S) of the wetted surface, is independent of the pressure, and may be expressed by the term 5 . F(v)- An obvious error in this assumption is that v is the mean velocity of the stream and not the velocity of the stream- lines along the bed and sides of the channel. In practice, however, the errors in the formulae based upon these imperfect hypotheses are largely neutralized by giving suitable values to the coefficient of friction (/). When a constant volume (Q) of water feeds a channel of given form, the water assumes a definite depth, a permanent regime is said to be established and the flow is steady. If the transverse sectional area (A) is also constant, then, since Q vA, the velocity v is constant from section to section and the flow is said to be uniform. Usually the sectional area A is variable and therefore the velocity v also varies, so that the motion is steady with a varying velocity. Any convenient short stretch of a channel, free from obstructions, may be selected and treated, without error of practical importance, as being of a uniform sectional area equal to that of the mean section for the whole length under consideration. 2. Steady Flow in Channels of Constant Section (A). The flow is evidently uniform; and since A is constant, the STEADY FLOW IN CHANNELS. depth of the water is also constant, so that the water-surface ^^ft .. T is parallel to the channel-bed. ; \G Consider a portion of the ^ feg^ stream, of length /, between the two transverse sections ^*^^^^^^S^::M^^ Let / be the inclination of the bed (or water-surface) FlG> I25 ' to the horizon. Let P be the length of the wetted perimeter of a cross- section. Then, since the motion is uniform, the external forces act- ing upon the mass between aa and bb in the direction of motion must be in equilibrium. These forces are : (1) The component of the weight of the mass, viz., li'Al sin /' = iv A It = ivAlj = wAk, h being the fall of lex r el in the length /. NOTE. When / is small, as is usually the case in streams, h = tan * = sin t = /, approximately. (2) The pressures upon the areas aa and bb, which evidently neutralize each other. (3) The frictional resistance developed by the sides and bed, viz., Hence wAh - PlF(v) = o, or F(v) _Ah _ 10 = ~Pl ~ m *' m being the hydraulic mean depth. STEADY FLO IV IN CHANNELS. 223 It now remains to determine the form of the function F(v). In ordinary English practice it is usual to take W 2g* /"being the coefficient of friction. Then f mi. J /2g - A/ ~~ ^mi = c V mi, or C being a coefficient whose value depends upon the roughness .of the channel surface and upon the form of its transverse section. The total head H in a stream is made up of two parts, the one being utilized in producing the velocity of flow and the other being absorbed in frictional resistance. Thus 2g ' 111 W In long channels and in rivers in which the slope of the bed ? does not exceed 3 ft. per mile the term - - is very small as 2 <~ compared with - - and may be disregarded without sensi- ble error. In this case m w Ex. i. A channel of regular trapezoidal section, with banks sloping at 30 to the vertical, has a bottom width of 8 ft., and a width of 16 ft. at the free surface. It conveys 288 cu. ft. of water per sec., and the fall is i in 2000. Find the mean depth, the mean velocity of flow, and the coefficients /"and c. 224 EXAMPLES. Depth of waterway = 4 tan 60 = 4 ^3 ft. A = ^-(8 + 16)44/3"= 48V3~sq. ft. P = 8 + 2 x 4 4/3 sec 30 = 24 ft. Therefore the mean depth = - = 24/3 = 3.464 ft. o oo The mean velocity of flow = = =,3^3"= 3.464 ft. per sec. 45V 3 Hence ' J 2000 2000 Therefore / = .009237 and c 82.63. Ex. 2. How much water is conveyed away by a horizontal trench 10 ft. wide, the depth of the water at entrance being 5 ft., and the sur- face falling i ft. in 2400 feet? (Take/ = .008.) Area at upper end = 50 sq. ft. ; at lower end = 40 sq. ft. ; the mean area = (40 + 50) = 45 sq. ft. Therefore, if Q cu. ft. are conveyed, = velocity at upper end ; = that at lower end ; and mean velocity = . 45 The wetted perimeter = 20 ft. at ^upper end ; = 1 8 ft. at low er end ; and mean wetted perimeter = - (20 + 18) = 19 ft. Thus the hydraulic mean depth m . Hence ! + (2.Y-L = (-2.V - 8 x 2400 /gyj^ \So) 64 \4o) 64 f| \45y 6 4' and Q = 389 cu. ft. per sec. 3. Retarding Effect of Air, etc. The retarding effect of the air upon the free surface of a river or of the water in a canal or in any channel has not yet been accurately determined. It may be assumed that the resistance per unit of free surface RETARDING EFFECT OF AIR, ETC. 225 due to the air is about one tenth of the resistance due to similar unks at the bottom and sides of smooth channels. Thus if JC is the width of the free surface in a smooth channel, the wetted perimeter becomes P -| . In general, the wetted perimeter may be expressed in the form P -f- , ft being 10 for smooth channels and greater than 10 for rough channels. The value of/? is evidently diminished by opposing winds and increased by following winds. Again, in the formula . F(v) mi = - , W m f= jy] and i l= -j\ are similarly related in the deter- mination of 7>, the mean velocity of flow. If v is constant, the product mi must also be constant, so that if m increases i must diminish, and vice versa. Thus in a very flat country the flow may be maintained by making m sufficiently large, while, again, if the channel-bed is steep m is small. The erosion caused by a watercourse increases with the rapidity of flow. At the same time the sectional area (A) of the waterway also increases, so that the velocity of flow v diminishes. Thus there is a tendency to approximate to a <* permanent regime " when the resistance to erosion balances the tendency to scour. Hence, throughout any long stretch of a river passing through a specific soil, the mean velocity of flow will be very nearly constant if the amount of flow (Q) does not vary. Generally speaking, the volume conveyed by a river increases from source to mouth on account of the additions received from tributaries, etc. Since Q increases, A must also increase; and if mi or v is to remain constant, i must diminish. It is to be observed that the surface slopes of large rivers diminish gradually from source to mouth. 226 RETARDING EFFECT OF AIR, ETC. For a given discharge (Q) the mean depth (m) diminishes, as i increases, and, as the cost of constructing a canal is approximately proportional to the mean depth, it is advisable to give the bed as large a slope as possible. But the velocity of flow (v) also increases with i, and the slope must therefore not exceed that for which v would be so great as to cause the erosion of the banks. On the other hand, v must not be so small as to allow of the growth of aquatic plants or of the deposition of sand, gravel, and other detritus, which would soon obstruct the waterway and add a considerable item to the cost of maintenance. Between these extreme limits the slope may be varied in any required manner, the controlling influ- ences being the configuration of the ground and the nature of the soil through which the canal passes. In every case a careful determination should be made of the best combination of the three elements 7', /, and A which would give a specified discharge. In France the canal beds have slopes varying frorn I J to 20 in 10,000, and the magnitudes of both v and t may be considerable when the canal passes through rock or through a well-compacted material capable of resisting. erosion. According to Belgrand the value of v for water carrying fine particles of loam should exceed I ft. (.25 m.) per secondhand should not be less than 2 ft. (.5 m.) per second if the waters are laden with coarse particles v of loam or sand. In clear water, the growth of weeds, etc., which would seriously inter- fere with the flow, is prevented if the velocity of flow is from 2 to 3 ft. (.5 m. to .8 m.) per second. The slope of an aqueduct, in which no trouble is to be anticipated from plant-growth, may be as small as 3 in 10,000, and may even fall to I in 10,000 when the waters are excep- tionally clear, as in the case of the aqueducts on the Dhuis and Vanne. On the other hand, the slope should rarely, if ever, exceed 12 in 10,000, and as a general rule the slope should be less than 10 in 10,000. The ordinary channel formula, viz., u = c Vini y is applicable to the flow in a conduit, so long as the RETARDING EFFECT OF AIR, ETC. 22J conduit does not run full, and since v is proportional to Vm it is. a maximum for some definite depth of water. When the water fills the conduit, the formula for channel-flow ought to> change suddenly so as to agree with that for pipe-flow, and in this respect the theory is therefore imperfect. The mean velocity of flow in a conduit should not be less than about 2 ft. (0.5 m.) per second, and may be as great as 5f ft., (1.5 m.) per second. High velocities enable the waters to> carry off floating debris and sand particles. There should be no sudden changes of slope or of section, as they favor the formation of eddies and the deposition of detritus. The following table of slopes and mean velocities is taker* from the article by Daries in the Encycl. Sc. des Aide- Memoire : Slope in 10,000. Mean Velocity per Second. Nature of Canal Sides. Feet Metres. Craponne Canal Marseille " Carpentras " Saint-Martory Canal . Verdon Canal IO 3 to 7 2 to 4 5.4 i-5 to 2 2 2 tO I 2 1.2 3 5 2-5 2.1 Alluvial soil Earth and rock Vegetable soil, fis- sured limestone Clay Calcareous rocks Clay, pudding-stone^,, rock Alluvial soil Calcareous rocks Earth Millstone-grit mason- ry with a ff-inch (= .02 m.) facing in cement Millstone-grit mason- - ry with a ff-inch (= .02 m.) facing in cement Rubble masonry with a |f- in. (= .oism.) facing in cement Rubble masonry with a||-in. (= .015 m.> facing in cement .. 3-3 2.53 to 6.56 1.6 4 2-5 1.64 1. 21 1.6 7 I.3I 1.18 3-3 2-95 2.62 2.36 I .77 tO 2 5 .76 5 37 51 4 .36 i 9 .8 .72 Neste ' ' Beaucaire Canal. .... Dhuis Aqueduct Naples Aqueduct .... Montpellier Aqueduct Croton Aqueduct 228^ ON THE FORM OF THE SECTION OF A CHANNEL. 4. On the Form of the Section of a Channel. The funda- mental formulae governing the form of the transverse section of a channel are Q = Av and 7,2 __ # mi __ # t. Therefore, also, Pi* = c*Qi. For channels of the same slope r 2 oc ;//. Take v^ am, a being some constant. Then, if dv is a small change in the velocity corresponding to a small change dm in the hydraulic mean depth, 2v . dv = a . dm, and therefore dv dm i' ~ 2m Thus the hydraulic mean depth must be changed 20 per cent to produce a change of 10 per cent in the velocity. Again, Q a Pi*. But P increases with Q, and therefore Q increases more rapidly than ?' 3 . For example, an increase in the velocity of less than 3-J- per cent will cause an increase of 10 per cent in the discharge. For channels giving the same discharge /V 3 oc i. For a given volume of water there must be a sensible change in the slope to produce an appreciable change in the ON THE FORM OF THE SECTION OF A CHANNEL. 229 velocity of flow, although, generally speaking, the wetted pe- rimeter (P) diminishes or increases as / increases or dimin- ishes, and thus v. and therefore -^, increases or diminishes yi more rapidly than i. An increase of 10 per cent in the ve- locity causes a diminution of about 4 per cent in the sectional area of the waterway. For channels of the same slope and giving the same dis- A* charge Pv 3 and also -rj are constant. A further condition is required before the sectional area can be determined. PROBLEM I. A canal of rectangular section and of width x is to convey water of depth y with the condition that either the sectional area (A) of the waterway is to be a constant quantity or the wetted perimeter (P) is to be a minimum. It is proposed to find the relation between x and y so w- ;A (/////Y//////////////Sj//////////////////< 1 FIG. 126. that (a) the velocity of flow may be a maximum, (b) the quantity of flow may be a maximum. and If v is a maximum., If Q is a maximum, P . dA - A . dP 3A 2 . PdA A* . dP 230 ON THE FORM OF THE SECTION OF A CHANNEL. In each case, if dA = o, i.e., if the area is constant, then and if dP o, i.e., if the wetted perimeter is a minimum, then dA = o. Thus the same results are obtained for the problem in its different conditions. Now A = xy and P = x -f- 2y. Therefore dA = y . dx -f- x . dy = o, and dP = dx -f- 2 . dy = o. Hence x Therefore, also, A x XI and 2 V 2 A suitable value for c corresponding to the slope i or to the value of m f = ) can be obtained from the Tables of Bazin, Kutter, or Manning at the end of the chapter. ON THE FORM OF THE SECTION OF A CHANNEL. 231 PROBLEM II. The section is usually in the form of a quadrilateral, the non-parallel sides sloping at an angle, #, FIG. 127. depending upon the nature of the soil through which the channel passes. For example, in a canal with retaining walls 6 = 63 36', with stiff earthen sides, faced, = 45, with stiff earthen sides, unfaced, =. 33 41', with sides in light or sandy soils = 26 34'. In such a channel let x be the bottom width and y the depth of the water. Then, the remaining conditions being the same as those in Problem I, it again follows that dA o and dP = o. But A y(x -f- y cot ff) and P = x -f- 2y cosec 0. First. If is given, dA = o = y . dx + (x + 2 y cot and dP = o == "dx -f- 2 cosec . dy. Therefore 2y cosec = x -^ 2y cot 0, 23 2 CW THE FORM OF THE SECTION OF A CHANNEL. or or x sin 6 = 27(1 ~~ cos #), tan - = . 2 2 The section may be easily sketched, as in Figs. 128 and 129. A C B FIG. 128. From the middle point C of AB, the bottom width, draw CF at right angles to AB and equal in length to the depth of" the water. Then AB CF= 2t 2> being the given slope of the sides. With F as centre and FC as radius describe a circle. From the points A and B draw tangents to touch this circle at D and E. FA evidently bisects the angle CAD. Therefore CAD CF CF B tan - - = tan CAF -^ ^ cot -. 2 AC ^AB 2 Hence n CAD = 6, and AD, BE have the slope required. Again, 2 cos B sin B ' ON THE FORM OF THE SECTION OF A CHANNEL. 233. or VA sin 8 2 - COS B and 1 cos , />= 27-2^3-+ 2, cosec* 2 COS # 2y2 = 2 '-ifirF- =7- Therefore A y and the coefficient c being obtained from the tables. The following Table gives the best relative values, per unit cf area, of x, y, m, and P, corresponding to specified values of 0, and the actual values may be obtained by multiplying those of the Table by VA: 9 x y m P 90 I.4I4 .707 -3535 2.828 _6o_ .877 .760 .380 2.633 45 .613 .740 .370 2.706 4 .525 .722 .361 2.772 36 52' .471 .707 -3535 2.828 35 -439 .697 .3485 2.870 30 . .336 .664 .332 3.012 ^26 34' .300 .636 .318 3.144 The above values cannot always be exactly adopted in actual practice. The character of the soil, the importance of preventing excessive filtration, and the difficulties of construe- 234 ON THE FORM OF THE SECTION OF A CHANNEL tion and maintenance, often render it necessary to insure that the depth of the water shall not exceed a certain limit, say 8 to 12 ft. (2 m. to 3 m.). In France the depth of irrigation- canals is between 4 and 6^ ft. (1.2 m. and 2m.). Second. If the bottom width x is fixed, then dA = o = (x -\- 2y cot &)dy y 2 - cosec 2 & . dO and dP = o = 2 cosec 6 . dy 2y . 2 ^ . dO. Hence x -\- 2y cot y 2 cosec 2 cos 0' or x sin cos j(2 cos 2 or and therefore sin 20 a x y cos 20, tan (?r 20) = -- tan 20 = . It may be observed that as the width (x) of the bottom increases, also increases. If the width is nil, then tan 20 = oo and = 45, so that the triangular section of minimum perimeter is a semi-square. Third. If the depth y is fixed, then dA = o = ydx y 2 cosec 2 . d& and dP = O = dx - 2J/ . dO. J sin 2 Therefore or cos = i and = 60. ON THE FORM OF THE SECTION OF A CHANNEL. 235 PROBLEM III. To find the proper sectional form of a channel of bottom width 2a so that the mean velocity of flow may be constant for all depths of water. Let x, j/, Fig. 1 30, be the co-ordinates of any point P in the profile referred to the middle point O of AB, the bottom width, as origin, and let s be the length of AP. FIG. 130. Since v is to be constant, m must also be constant, and therefore A fy- dx -p: = = a const. m, P s + a which may be written / y .dx = m(s -f- ). Differentiating, y . dx m . ds m(dx* and therefore dx __ dy Integrating, x - log^ (j/ -)- i/j/ 2 m 2 ) -\- Cj c being a constant of integration. But y = a when x o, and . . o = \og e (a -f- Va 2 236 ON THE FORM OF THE SECTION CF A CHANNEL. Hence, too, Adding together the last two equations, * m* -*- 2y be + -e "', or y which is the equation to the required profile, and is a curve which belongs to the class of catenaries and which evidently flattens out very rapidly. If the bottom width is such that a = m b, the equation becomes and the profile is a true catenary of parameter m, with its axis coincident with the bottom and v its directrix coincident with the vertical at the middle of the section. FIG. 131. PROBLEM IV. A channel of given slope has a given sur- face width AC, vertical sides AB (= j^) and CD (= j/ 2 ) of given depths, and a curved bed BD (= L) of given length. ON THE FORM OF THE SECTfOM OF A CHANNEL 237 The amount and velocity of flow in the channel will be a maximum when the form of the bed BD is a circular arc. This can be easily proved as follows : Since the slope is constant, v oc Vm oc But P ( L -\- y l + y^ is a constant quantity, and there- fore v and also Q will be a. maximum when A is a maximum. Hence, too, the area between the chord BD and the curve must be a maximum, and therefore the curve must be a circular arc. The proof of this by the Calculus of Variations is as follows : Take O in CA produced as the origin, OC as the axis of ,#, and the vertical through O as the axis of/. Then A = I ** ydx is to be a maximum. t/JCl dy is a given quantity, OA being = x^ , OC = * 2 , and -- Let V ' = y -\- a V / i -|- / 2 , a being some constant. Then / * V . dx is to be a maximum, tJx l and therefore dV v = *Tp + c - that is, and thus 238 CW THE FORM OF THE SECTION OF A CHANNEL. Therefore dx i c, v Integrating, P Va* (^ y = W- (c.-y?, the equation to a circle of radius a. Hence the profile BD is a circular arc. The maximum depth of the channel is c l a. The constants c l , c 9 , a can be found from the three condi- tions that the arc is of given length and has to pass through the two fixed points B and D. PROBLEM V. The Semicircular Channel. Theoretically > the best form of channel for a given waterway is one in which the bed is a circular arc (Prob. IV), as the wetted perimeter is then a minimum and the mean depth (or radius) a FIG. 132. maximum. In the semicircular channel, Fig. 132, let the free surface subtend an angle at the centre. Then A = ( e _ sin )(, -5S. I /3 V ^ and r being the radius. Therefore A r m = = - sin 6 ON THE FORM OF THE SECTION OF A CHANNEL. <2 2 Hence, since mi = by* = b~, A If the channel runs full, TT, and then As a first approximation it may be assumed that for small channel sections with cement faces ...... b = .00022 " channels of mean dimensions with smooth faces b = .00017 <4 channels of large dimensions ................ b = .0001 1 In metric measure these coefficients become .0004, .0003, and .0002, respectively. Miscellaneous Problems. The bed of the aqueduct at Naples is semi-elliptic, but beds in the form of a semi-ellipse, a cycloid, a parabola, or an hyperbola, would only be adopted under very exceptional conditions, as when a curved profile is required with a limited depth. The waterway and the wetted perimeter can, of course, be approximately calculated from the known properties of these curves. For the semi-elliptic section, if a and b are the semi-major and minor axes, ab A = 7t~, and fwhere t/= i. 1.0025, 1.01, 1.0226, 1.0404, 1.0635, 1.0922, 1.1267. I - 1 677, 1.2155;} 1 when ~^- 6 = i -i. - 2 - -3, -4, -5, -6, .7. -8, -9- J For the cycloidal section, if r is the radius of the generating circle, A = 7tr* and P = Sr. 240 AQUEDUCTS. Therefore and the flow equation becomes If the water-line is at y2^4, defined by the angle # which the radius OA of the generating circle makes with the vertical, then - 2r 2 (i cos 0)(7T # + sin 6 1 ) = ;r _ + 2 sin + 27i cos - 20 cos + and 8 P = Srcos -. 5. Aqueducts. The aqueduct of the ancients was of rectangular section and was sometimes of very large dimen- sions as compared with the volume of water to be conveyed. Although in modern times there are examples of rectangular sections, it is now more usual to make them circular, egg- shaped, square with a diagonal vertical, or trapezoidal.. Aqueducts are also constructed of forms which are combina- tions of the circle and egg-shaped, or of the trapezoid and circle. When a mean volume of water is to be conveyed and when provision has to be made for a definite height, as, for example, for a man standing upright, preference is given to the egg- shaped aqueduct. In the sections shown by Figs. 133 to 137 it will be observed that a rise of the water-line near the top causes an AQUEDUCTS. 241 appreciable increase in the wetted perimeter, while there is no proportional increase in the waterway. Thus the mean depth (m) and therefore also the mean velocity (v) of flow continually diminish. The a priori conclusion may be drawn that the discharge (Q) is not a maximum when the pipe runs full, but when the water-line is some distance below the top. The FIG. 133. FIG. 134. FIG. 135. FIG. 136. FIG. 137. differential equation defining this position may be easily found as follows (Prob. i, p. 229): Therefore A^i, ~pi>' Since Q is to be a maximum, Therefore .dA-A*.dP or is the equation required. 242 AQUEDUCTS. If the velocity of flow is to be a maximum, dv = o, and therefore IA\ P .dA - A.dP dm = o = a\-^l = - Fir, , or P . dA- A . dP = o. Ex. I. Circular Section. Let the wetted perimeter sub- tend an angle B at the centre. Then r r* . dB - id sin#) and dA= (i cos#); 2 V 2 V P = rB and dP = r . dB. Hence for a maximum discharge 2 tf_ 3 ^ C os#+sin0 = o. 308 is the value of B satisfying this equation. For a maximum velocity -dO . B(i cos 0) (6 - sin 0) = O, 2 2 or (9 = tan By and =257 27' is the value of B which satisfies this equation. In circular aqueducts the angle B is usually about 240, which insures a certain clear space above the water-line. Then, also, P = 4.189;-; ^=2.528^; ;// = .6r. AQUEDUCTS. 243 EXAMPLE 2. A Square Section with Vertical Diagonal* -i-Let a side of the square -- a, and let x be the length of the portion of the side which is not w r etted. Then and dA x . dx ; T-J and dP = - 2 . dx. FlG - Hence for a maximum discharge i x %\ + 2(0* jdx = O, or $X* I2^r -f- 20* = O. Therefore = -(6- ^26)= .I8, and the depth below the apex of the water-line = -= = .1274^. For a maximum velocity of flow or x z ^.ax -f- 2a* = p, 244 AQUEDUCTS. and therefore x = a(2 t/2*) = .58580, and the depth of the water-line below the apex = = = .41420. V2 EXAMPLE 3. Egg-shaped Section. This form of aqueduct consists essentially of three parts, a lower portion bounded by a semicircle of radius r l , an upper portion bounded by a cir- FIG. 140. cular arc of lesser radius r 2 , and an intermediate portion bounded by circular arcs of radius r 3 , which meet the lower and upper arcs tangentially. The depth of the intermediate portion is defined by the angle a which the radius O 3 O 2 makes with the horizontal, and the position of the water-line AA is defined by the angle & which 2 A makes with 3 O 2 produced. Then AQUEDUCTS. 245 If the water-line is above BB, and If the water-line coincides with BB, 6 = 0, and then nr 2 r 2 -4 = ^ + r 3 2 - (r s - ^)( r 3~ r a) sin + ^ sin 2 and P = ^r x -f- 2r 3 <*. If ^ is the vertical distance between O l and the highest point, z = r 2 -\- (r 3 r t ) sin <*. Also, If the water-line CC is below ^9j5, let be the angle sub- tended at <9 3 by the arc .567, and let 6> 2 dT = x. Then, since is now <9, ^r sin (6 0) = (r 3 r 2 ) sin and x cos (# a) = r 3 cos ( 0) (r s rj), two equations giving x and in terms of 6 and the radii. The area of the waterway is now the area up to BB diminished by the area of the slice between BB and CC, and this area r 2 x* = sin 2a + r 3 2 r 3 (r 3 r a ) sin -| -- sin 2(# ). 2 2 246 AQUEDUCTS. Hence nr \ I I r 2 A=-- + rf*- (r 3 - ^)(r s - r 2 ) sin-ar + -- sin { r 2 sin 2 -j- r 3 2 r 3 (r 3 r 2 ) sin -[- ' sin 2(6 a] 2 r,) sin ^ r, sin sin 2(^ n). and The larger diameter is usually at the bottom for aqueducts, but almost invariably at the top for sewers. The discharge for sewers may be calculated by Bazin's formula, but an allowance of 20 per cent should be made in order to make provision for deposits and, where they occur, for water-pipes, electric conduits, etc. Care should also be taken that the section is sufficient to carry away the water from the heaviest rains and from the branch drains in such manner that the water in the sewer does not rise above a certain level. Assuming that the time of flow in the sewer is three times that of the rainfall and that the maximum downfall is 27.5 gallons (=125 litres) per second, Belgrand has proposed for the discharge of the Paris sewers the formula 5 X .188 = A Vmi, S being the drainage area in acres. In metric measure, 5 being the drainage area in hectares, 5 X .0239 = A Vmi. In branch drains and in smaller systems the influx of water is much more rapid and the time of flow should not be estimated at more than twice the duration of the rainfall. FORMUL/E OF PRONY, EYTELH/EIN, ETC. 247 NOTE. In designing sections for open channels or aque- ducts, complicated preliminary calculations may be generally avoided by employing a graphical method. Selecting a pro- visional section, the water areas and wetted perimeters may be obtained for different depths of water and the corresponding mean depths plotted to any convenient scale. Repeating these operations for different sections, the mean-depth curves will quickly indicate the best section to be adopted. 6. Formulae of Prony, Eytelwein, Beardmore and Tadini. A careful study of Chezy's experiment on the Courpalet cut (Orleans canal) and of twenty-three experiments made by Dubuat on wooden channels of small section, led Prony, in 1804, to adopt the equation F(v) = av -4- bv* 1 = mi. w in which = 22472.5 and -r- = 10607.02. About the year 1815, Eytelwein, taking into account sixty additional experiments on the Rhine and Weser by Woltmann, Funk and Brunings, proposed slightly different values for a and b, viz., i = 4l2II.H and ^=8975.43- The expression mi has the same value with Prony 's as with Eytelwein's coefficients when the velocity is about 72 ft. per minute, and for a small change in this velocity the variation in the value of mi is also small and of little practical importance. For other velocities the value of mi with Prony 's coefficients will be greater or less than the value with Eytelwein's coeffi- cients according as the velocity of flow is greater or less than 72 ft. per minute. The formula with Eytelwein's coefficients was for a long 248 FORMUL/E OF PRONY, EYTELWEIN, ETC. time used by engineers, and was preferred as giving the most reliable results. For values of v exceeding 20 ft. per minute the term av is small as compared with bv*, and may be disregarded without much error, This formula then becomes bv* = mi, and therefore, according to Prony, v = = Vmi =103 Vmi y V ' b and according to Eytelwein, v = = Vmi = 95 Vmi. Intermediate between these is Beardmore's formula, viz., v = 100 \'mi. Barre de St. Venant has suggested the relation mi .000136^" (or mi = .OOO4^ TT , if a metre is the unit). The above formulae, now obsolete, involve a grave error, as it is assumed that the resistance due to the roughness of the wetted surface is a constant quantity. Bazin's experiments have clearly shown that the resistance may vary between very wide limits depending upon the nature of the materials and soil which form the bed and sides of* the channel. For a deep and wide channel, in which the slope of the bed is small, approxi- mately accurate results are given by Tadini's formula, ?/ = 91 Vmi (or v = 50 \/mi, if a metre is the unit). BAZIWS FORMULAE. 249 7. Bazin's Formulae. Between 1855 and 1859, Darcy and Bazin carried out a number of experiments in a cut leading from the Bourgogne canal. The channel sections were of different forms and dimensions, the sides were faced with wood, cement, hewn ashlar, bricks, rubble masonry, and earth, and the slope of the beds varied from .001 to .10. The results, for the rectangular and trapezoidal sections, sensibly agreed with the calculations obtained from the formula "but with circular and egg-shaped sections the calculated are about 10 per cent less than the actual results. In practice it is most convenient to take v = ~ I/mi = c where b = = a -I --- . c 2 r m (Y and ft are not constant, but have values depending upon the character of the channel faces and bed. Bazin gives the following table: Character of the Wetted Surface. Value of a, the Unit being Value of |8. A Foot. A Metre. Smooth cement, planed wood, etc Cut masonry bricks planks . .. .000046 .000058 .000073 .000085 .OOOI2 .00015 .00019 .00024 .00028 .00040 .0000045 .0000133 .OOOO6 .00035 .0007 Rubble masonry Earth . Boulders ( Kutter) Tables at the end of the chapter give the values of the coefficients b and c y a metre being the unit. Reviewing the results of more than 700 experiments carried out in France, Europe, the United States, and British India, 250 GANGUILLET AND KUTTEKS FORMULA. etc., upon canals and rectangular, trapezoidal, semicircular, and circular aqueducts, of different dimensions, Bazin, in 1897, (Ann. des Fonts et Chaussees^) deduced the formula I57 - 6 87 for v = - Vmi, if a metre is the unit). X + -^= 1/m This equation, again, may be most conveniently written in the form v = c ^mi, and Tables at the end of the chapter give the values of c for the six different classes into which Bazin has divided all channels, the corresponding values of the coefficient y being given by the following table: Class. Character of the Wetted Surface. y, the Unit being A Foot. A Metre, 1. II. III. IV. V. VI. .109 .290 833 1.540 2-355 3.170 .06 .16 .46 85 I 30 1.7 i Earthen channels or rivers, presenting exceptional resistance ; the beds covered with boulders and 8. Ganguillet and Kutter's Formula. Bazin's is the only formula used in France, but in England, Germany, and the United States engineers prefer the formula of Ganguillet and Kutter, viz., v = c ^mi, the value of c being given in a Table at the end of the chapter. GANGUILLET AND KUTTERS FORMULA. Also the coefficient 251 c = a, 1, and p being certain constants and n a coefficient depend- ing only on the roughness of the channel sides and bed. If the unit is &foot, a=4i.6; 1 1.8112; p= .00281. If the unit is a metre, a=23; 1= i; p = .00155. The unit being a foot, n varies from .008 to .05 and the following table gives the values of n which will be found of most use in practice: Character of Sides. n. Authority. ooo -] Smooth cement .01 A mixture of 2 of cement to I of sand Rough planks .. Ashlar or brickwork ... OTO Canvas on frames .<-! j OI Rubble masorrv OI7 Rivers and channels in very firm gravel. ... - . . .02 Ganguillet " " perfect order, free from 'de- tritus (stones, weeds, etc.). " moderately good order, not quite free from detritus or weeds . . .025 o^ and Kutter " " bad order, with weeds and detritus QOC OC Canals in earth above the average order Q22C -x " " " in fair order ,02 s ; " " " below the average order .027^ J- Jackson " " " in rather bad order, overgrown with weeds and covered with detritus. . 03 The difficulty of properly selecting the value of n is due to the fact that there is no absolute measure of the roughness of channel beds. In obtaining the above results Ganguillet and Kutter made a careful study of: 25 2 FORMULA OF MANNING, TUTTON, ETC. (a) The Experiments of Darcy and Bazin. These show that c depends both upon the roughness and the sectional dimensions. The values of OL and ft in Bazin 's formula vary with the character of the channel sides and bed ; but while in small channels the influence upon the flow of differences of roughness must be very great, it is certain that this influence diminishes as the sectional area increases, and that it will be nil when the area is infinitely great. (<) The Measurements of Humphreys and Abbot on the Mississippi, a stream of very large sectional area with a bed of very small slope. (c) Their own Gaugings in the regulated channels of certain Swiss torrents with exceptionally steep slopes and running through extremely rough channels. (d) The Effect of the Slope. The coefficient c diminishes as the slope, /, increases. The value of c does not vary much with the slope of the bed in small rivers, but in large rivers with small slopes the variation is considerable. 9. Formulae of Manning, Tutton, Humphreys and Abbot, and Gauckler. In 1 890, Manning proposed the formula v = cjvfi.fi = - #****, if the unit is a foot, or v = cjrifi* = m^i^y if the unit is a metre. In this formula, which gives good results, the coefficient n has the same value as the n in Kutter's formula. Bazin's and Manning's formulae are identical if c \'mi = ~,-r Vmi = ^fftb'}, i.e., if , = -L = ^. EXAMPLES. 253; By an independent method, Tutton, in 1893, deduced the corresponding formula, n being again the same as in Kutter's formula. As a result of observations on the Mississippi in 1865, Humphreys and Abbot deduced the rather complicated formula v {3 -873 (*'**)* -0388[ 3 , the unit being a foot, v = \(6gm'i^ .O2I4J 2 , the unit being a metre. or In this expression, which is of especial value for large watercourses, m' is the ratio of the sectional area to the total perimeter. Gauckler's formulae for canals, v = cm*i, if the slope is < 7 per I oooo, and v = c'mh't, " " > 7 per 10000, and Hagen's formula, v 2. 4 3 ***, the unit in each case being a metre, have not been used in practice. Ex. i. A channel with a fall of i in 10,000 has brickwork faces, is of rectangular section, 20 ft. wide, and is to convey 200 cu. ft. of water per second. What must be the depth of the water ? Let x be the required depth. Then A = 20.*; P = 20 + 2,v, IO.T and m = - 10 + x 2. 54 EXAMPLES. Also, 200 20JT 10 ./ 10* I C ./ \VX = if = c {/ . = !/ . X 10 + X lOOOO 100 V 10 + X , This equation can be best solved by trial. Let x . = 5 ft. Then ' =^r = 3-33 ft-, ;and the Tables give 127.2, as the corresponding value of c. Therefore v = loo" i/3 ' 333 = 2 ' 3223 ft< per sec " and Q = iooz/ = 232.23 cu. ft. per sec., which is too great. Let x = 4 ft. Then m = = 2 8 c 14 and the Tables give 126.4 as tlie corresponding value of c. Therefore 126.4 A/ 40 It = I/ - = 2.136; ft. I /-w-v ' T 1 / ^ per sec. loo ' 14 and Q = Sot/ = 170.92 cu. ft. per sec., which is too small. Thus x must lie between 4 and 5 ft. Try x = 4.5 ft. Then . 14.5 and the corresponding value of r is 126.8. Therefore 126.8 .790 v = \ ' = 2.2338 ft. per sec., loo r 29 and <2 = 90^7 = 201.042 cu. ft., which is very nearly correct. By further trials the depth can be ob- tained within a fraction of an inch. EXAMPLES. 255 Ex. 2. A canal in earth with sides sloping at 40 is to convey 100 cu, ft. of water per sec., at a velocity of i ft. per second. What is the fall of the canal, and what are its most suitable dimensions? A = ioo. Then (see Table, p. 233), bottom width = .525 j/ioo 5.25 ft., depth of water = .722 4/100 = 7.22 ft., mean depth, m .361 | 7 ioo = 3.61 ft. By the Tables the corresponding value of c is 93.3. Therefore = 93-3 4/3- 6| and 3H24 Ex. 3. A length of the La Roche cut is in compact rock. Its bottom width is 0.70 m., the depth of the water is 0.50 m., one bank is vertical and the other slopes at 26 34' to the vertical. If the fall is i in 500, find the mean velocity and quantity of flow. The width of section at the surface = .70 + .50 tan 26 34' = o m .95. A -(.95 + .70). 50 = 0.4125 sq. m. P .50 -f .70 + .50 sec. 26 34' = 1.759 m - Therefore .4125 1-759 and the corresponding value of b in the Tables is .0423. Hence .0423 x TV = y .2345 x = .02165, and v = .512 m. per sec. Therefore, also, Q = .512 x .4125 = .2112 c.m. per sec. Again, using Bazin's formula for the filament of max. vel., z'max. = v + 14 |/;///= .512 + 14 x .02165 = o m .8i5 per sec., and -vb bottom velocity = (2/ ma x.) = o m .489 per sec. 256 EXAMPLES. Ex. 4. In another length of La Roche cut, in earth, the banks slope at 45, the bottom width is 0.3 m., and the depth of the water is 0.5 m^ Find the coefficients b and c, the discharge being .21 12 cu. ft. per second, and the fall i in 500. A = -(1.3 + -5). 5 = o-4 sq. m. P = .3 + 2 \'7$ = r n . 7 14- Therefore m = ~ - = .23337. 1.714 -> j j 2 Also, v = - = o m .528 per sec. 4 Hence = c x .02162 = x .02162 \'b and c 24.4, b = .00168, which closely agree with the results given by the Tables. Ex. 5. Find the quantity of water conveyed by a channel of trape- zoidal section lined with brickwork and having a fall of 6 in looo. The water-surface width is 7.185 ft., the bottom width is 6.56 ft., and the depth of the water is 4.92 feet. A = (7.185 + 6.56) x 4.92 = 33.813 sq. ft., P = 6. 56 + 2 x 4.954 = 16.468 ft. Therefore m = fgf = 2.053. Hence Q = Av = c x 33.8I3J/ 2.053 x ---- = c x 3.7528. For m = 2.053 Baziri's Tables give c = 124.6, and then Q = 467.6 cu. ft. per sec. Manning's " " c = 128.6, and then <2 482.6 " " Kutter's " " c 130.4, and then Q 489.36 " " and the differences in the three cases are not considerable. VELOCITY VARIATION IN TRANS YERSE SECTION. 257 10. Variation of Velocity in the Transverse Section of a Watercourse. The discharge (Q) across any transverse sec- tion of a watercourse is the product of the area (A) of the section and the mean velocity (v) of flow. Thus Q=Av. The value of v for channels of small section can easily be found by discharging into a suitable reservoir for a definite interval of time, when Q can be estimated ; and since A is known, v can be at once calculated. This method is imprac- ticable with watercourses of large dimensions. The profile of the section must then be carefully plotted, when its area can be obtained with a planimeter or by the method of mean heights. The velocity of flow varies from point to point throughout the section in a most irregular manner, and its value has not been fixed by any single law. By using a meter or gauge the velocity may be measured at a large number of points, and in this manner the mean velocity (v} and the maximum velocity (s> max .) can be very approximately determined. The velocity, however, varies so much and depends so largely upon the conditions under which the flow takes place, that it seems hopeless to expect that the compli- cated law of velocity distribution can be expressed in a general formula. The numerous experiments of Bazin on the Bour- gogne canal and on the Seine and Saone, of Cunningham on the Ganges canal, and of Humphreys and Abbot on the Mississippi, all go to prove this and at the same time throw much light upon the whole subject. It has been shown that the ratio - diminishes as the resistance of the sides and bed, which is measured by the expression , increases. The ratio, for example, is about .85 in a channel with a very smooth surface and falls to about . 50 when the channel is cut through 258 VELOCITY VARIATION IN TRANSVERSE SECTION. earth. As the surface resistance diminishes the value of ^ tends to become very small and ultimately zero, while the ratio - tends to become unity. Bazin therefore expressed the relation between v^.. and v in the form -+'(3) in which the function /M H vanishes with ,. \ v 2 1 v* A special case is Bazin 's empirical formula, (3) (4) the values of b and <: being given by the Tables, and K being a coefficient depending upon the form of the section and the conditions of flow. For example, if ^' max . is the maximum surface velocity for a given section, V for a watercourse of great width as compared with the depth and for a channel of restricted dimensions, as in ordinary practice. Again, if ^ max . is the maximum velocity for the whole sec- tion of such a channel, and if v m is the mean velocity along VELOCITY VARIATION IN TRANSVERSE SECTION. 259 the vertical in which the maximum velocity lies, then, approximately, (7) in which h is the depth of the water on the vertical in ques- tion. (If a metre is the unit, the values of K in the three last formulae are 20, 14, and 6, respectively.) For channels of mean dimensions Prony has suggested the formula *' 7-7% + ^max. , '~' (If the unit is a metre, substitute 2.37 for 7.78, and 3.15 for 10.34-) In the same case Dubuat gives (9) in which v b is the velocity at the bottom of the channel. For values of z>' max . up to about 11 or 12 ft. (3.5 m.) per v second the calculated values of the ratio vary but little from v the average value .8, a result which has been verified in certain special experiments. It is therefore considered sufficient to take and then, by eq. (9), When the water is of great depth the ratio -- falls to ^ max. 75, and to .60 if the bottom is covered with reeds. 260 VELOCITY VARIATION IN TRANSVERSE SECTION. Sonnet has theoretically deduced for watercourses of great width the relation so that if v = l^'max.* then v 6 = %v' msatm For a long time it was supposed that the maximum velocity (^max) was m the free surface, and its value was determined by observing the time in which floats passed between two trans- verse sections at a specified distance apart. Experiments have now demonstrated that this maximum velocity is at some point below, although in general near the free surface, and the floats will not give the proper value of the maximum velocity unless they are suitably submerged. It has also been found that the depth of this point of maximum velocity increases as the ratio of the width to the depth of the waterway diminishes, and may be as great as one third of the depth of the water. On any horizontal line at right angles to the axis of the channel the velocity diminishes with the depth of the water, is greatest towards the centre, and diminishes at an increasing rate on approaching the sides. The experiments of Darcy and Bazin have shown that the air-resistance is not the most important factor in causing the variation in the velocity throughout the section. With a gauge they determined the velocities at a number of points in the cross-section, and plotted the corresponding equal-velocity curves : (a) For a closed wooden pipe, of rectangular section, running full (Fig. 141); (/) For an open wooden channel running half full and formed by removing the upper side of the pipe in (a) (Fig. 142). The curves for the pipe are approximately rectangular and parallel to the sides of the pipe. The discharge in the open channel is slightly greater than one half of the pipe's discharge, VELOCITY VARIATION IN TRANSVERSE SECTION. 261 but there is no similarity between the equal-velocity curves in the two cases. In the open channel they become more -elliptical, tend to close at the centre, and cut the free surface obliquely, the angle of incidence becoming more and more acute towards the centre. The curves are also at a greater distance from the centre than the corresponding curves in the pipe. This very marked modification in the form of the FIG. 141. FIG. 142. velocity curves is due especially, in Bazin's opinion, to the absence of the upper boundary and to the consequent practical impossibility of an absolutely constant cross-section. Eddies and other irregular movements are .produced in the surface and give rise to corresponding losses of energy and velocity. Actual experiment, too, has shown that, even with a strong wind blowing down-stream, tending, as might be supposed, to cause an excessive surface velocity, the maximum velocity is still at some point below the free surface. For any given vertical in the section it appears to be approximately true that the velocity at about three fifths of the total depth is sensibly the mean velocity for the whole depth, and that the difference between the maximum and bottom velocities, viz., 7' max . v&j increases with the roughness and lies between i^ max . and <^ max> . In a semicircular channel of radius r the equal-velocity curves are circular, Fig. 143, and concentric with the bed, the 262 VELOCITY VARIATION IN TRANS I/ERSE SECTION. velocity v at the distance y from the centre being given by 38 Vn ' being the velocity at the centre. FIG. 143. Generally speaking, the equal-velocity curves are approxi- mately of the same form as the profile of the section (Figs. 143 to 146), and this is especially the case near the sides and FIG. 144. FIG. 146. bed. The curves at the bottom do not always reach the sur- face, but sometimes cut the sides. Again, experiments indicate that the law of velocity dis- tribution along any vertical in the section may be represented by a parabola of the 2d degree, with its axis horizontal and at the. same depth as the point of maximum velocity. Defontaine in an experiment on an arm of the Rhine deduced for the vertical at the centre of the current the analogous law u 4.8222 .066^, .... (13) u being the velocity at the depth y. (If the unit is a metre, u = 1.266 .252^.) VELOCITY VARIATION IN TRANSVERSE SECTION. 263 The following theoretical investigation of the velocity curve & based on the assumptions that : (a) The watercourse is of very great width as compared with the depth ; (b) The watercourse is of sensibly uniform depth ; (c) The fluid particles flow across a transverse section in sensibly parallel lines ; (d) A permanent regime has been established so that the pressure is distributed over the section in sensibly parallel lines ; (e) The resistance to the relative flow of consecutive fluid filaments is of the nature of a viscous resistance. Let Fig. 147 represent a portion of a vertical longitudinal 1 ct| I j >,! dy . j c *~- 1 -i > > B J 5 FIG. 147- section of the stream intersected by two transverse sections AB, CD, / being the distance between them. Consider a thin layer abed of thickness dy and width b, bounded by the sections AB, CD, and by the planes ad, be, at depths y and y + dy, respectively, below the free surface. The forces acting upon the layer in the direction of motion are: (1) The pressures on the ends ab, cd, which evidently neutralize each other. (2) The component of the weight = wbl.dy.sm i = wbli . dy; i being the slope of the bed. (3) The viscous resistances on the lateral faces of the layer under consideration. These are nil, since in a stream of indefinite width there will be no relative sliding between abed and the vertical faces on each side. 264 VELOCITY VARIATION IN TRANSVERSE SECTION. (4) The viscous resistances along the planes ad and be. The frictional resistance to distortion, i.e., to shearing, along such planes, is found to be proportional to the shear per unit of time, and is measured by the shear per unit of area at the actual rate of shearing. The coefficient of viscosity, or i ,, i shear per unit of area simply the viscosity, is the quotient : , shear per unit of time and defines that quality of the fluid in virtue of which it resists a change of shape. Adopting Navier's hypothesis, the viscous resistance along- ad = kbl , dy k being the coefficient of viscosity, and u the velocity at the depth y. The sign is negative as, since u increases with y, -j- is positive, and, at the same time, the action of the layers above ad is of the character of a retardation. The viscous resistance along be = kbl 4- kbl d(\ dy \dy) dy* dy^ Then, as the motion is uniform, wbli . dy - kbl- + kbl^ + kbl^dy = o. dy dy ' dy* ' Hence wi Integrating twice, wt (14) a and v s being constants of integration VELOCITY VARIATION IN TRANSVERSE SECTION. 265 It is evident that v s is the surface velocity, i.e., the value of #*when y = o. The equation may be written in the form 2WI wi 2~k Thus the velocity curve is a parabola ka having a horizontal axis at a depth F = . wi below the free surface. This is also the depth of the filament of maximum velocity idu \ I -j- and \dy / h^ W * v . + -,y*. (16) FIG. 148. Hence, by equations (14) and (16), w Let v m be the ' ' mean ' ' velocity for the whole depth h. Let z/i be the mid-depth velocity. Then . . . (18) and i lh 266 VELOCITY VARIATION IN TRANSVERSE SECTION. Hence wit? a result upon which Humphreys and Abbot have based a rapid method of gauging rivers. Let v 6 be the bottom velocity, i.e., the value of v when y = h. Then, by equation (17), w and therefore, wi ^max. - V b = -(k - Vy = N, SU PP OSe. . . (2 l) According to Bazin, ^ max . v 6 is sensibly constant and is approximately equal to 36. 3 Vki (=. 20 1// if a metre is the unit). Thus the general equation (15) of the velocity curve becomes = *W.-36.3y>**\j^-y-j. (22) This, known as Bazin 's formula, agrees well with the experiments on artificial channels and on the Saone, Seine, Garonne, and Rhine. It was found, in general, that -^^ = 1 . 1 7 in the Rhine at Basle and ranged from I . I to 1.13 in the other channels; 8 ran ^ ed from J 3 to 20; - F) VELOCITY VARIATION IN TRANSVERSE SECTION. 267 I - j F I . m -r = - in some artificial channels and in others ranged. from o to .2. These last results are not in accord with the Mississippi measurements. In the case of a rectangular channel of such width that the influence of the sides on the flow may be disregarded, th e mean radius, m, may be substituted for h and the mean velocity, v m , is sensibly the same as the mean velocity, v, for the whole section. Hence equation (22) may be written v or , (23) the value of b being given by the Tables. Filament of Maximum Velocity in the Surface. In this case Y o, and equation (21) becomes wi V max. ~ V6 = ^ (24) ^ 'max. being the value of z/ max . when the maximum velocity is in the surface. Equation (22) also becomes -. (25) Again, by equations (18) and (24), w 268 VELOCITY VARIATION IN TRANSVERSE SECTION. or ^'max.H- V* , V ~=- -- ....... (26) result already referred to. Boileau s Formula. Boileau assumes that the velocity o _______ v._ _____ M t _N_ curve is given by the equation Cy . . (27) . M above the point of maximum velocity, and below this point by the equation u=D-B? ..... (28) FIG. 149. When y = o, u = v t = A . When y=Y, D _ BY* = z/ max . = A - BY*+CY= v s - BY* + CY. Therefore Y Boileau 's experiments led him to infer that the difference D z> max . ( Y 2 ) is sensibly v constant. Designating this difference by d, so that D = ^ max . + d and B = pr 2 , Boileau 's equations become ^max. ^ + d y . . (29) representing the curve MM 2 , and = ^.+^~XF) ....... (30) representing the curve J^f r TABLES OF EROSION 4ND VISCOSITY. 269 ii. Tables of Erosion and Viscosity. -..TABLE INDICATING THE VELOCITIES ABOVE WHICH EROSION COMMENCES. Nature of the Channel Bed. z i V m i) b Met. Feet. Met. Feet. Met. Feet. if\ J 5 49 rR 3 ,fi . 20 3 f\n 9 2 3 Af\ 75 52 I.Q7 .40 r>f- I.5 1 3 1 .90 3- T 5 .70 2.30 _ r>9 Soft schist 1.52 5-OO 7 28 1.23 i 86 4-3 94 3.08 Stratified rocks . 49 T So 4.90 Hard rocks 75 427 2.27 7-45 U V *4 1U. J * TABLE OF VISCOSITIES (Everett's System of Units). WATER. MERCURY. Temp. (Cent.) Viscosity. Temp. (Cent.) Viscosity. Temp. (Cent.) Viscosity. Temp. (Cent.) Viscosity. O .Ol8l 35 .0073 .0169 3I5 .00918 5 .0154 40 .0067 10 .0162 340 .00897 10 0133 45 .0061 18 .0156 15 .OIT6 50 .0056 99 .0123 20 .OI02 60 .0047 154 .0109 25 .009! 80 .0036 197 .OI02 30 .008l 90 .0032 249 . 00964 The viscosity is given by ' ^cording to Meyer, and by 5212 '^ " .001 31, according to Slotte; 26 / being the temperature centigrade. 12. River-bends. The following explanation is due to Professor James Thomson (Inst. Mechl. Engs., 1879; Proc. Royal Soc. 1877). In rivers flowing in alluvial plains, the * N. B. The viscosities are in C. G. S. units, units and centigrade degrees multiply by 2.0481. To reduce to F. P. S. 270 RIYER-BENDS. curvature of the windings which already exist tends to increase owing to the scouring away of material from the outer bank and to the deposition of detritus along the inner bank. The sinuosities often increase until a loop is formed, with only a narrow isthmus of land between two encroaching banks of a river. Finally a cut-off occurs, a short passage for the water is opened through the isthmus, and the loop is separated from the river-course, taking the form of a horseshoe shaped lagoon or swamp. The ordinary supposition, that the water always tends to move forward in a straight line, rushing against the outer bank and wearing it away, and at the same time causing deposits at the inner bank, is correct, but it is very far from being a complete explanation of what takes place. When water flows round a circular curve under the action of gravity only, it takes a motion like that in a free vortex. Its velocity parallel to the axis of the stream is greater at the inner than at the outer side of the curve. Thus, too, the water in a river-bank flows more quickly along courses adjacent to the inner bank of the bend than FIG. 150. along courses adjacent to the outer. The water, in virtue of centrifugal force, presses outwards so that the water-surface of a transverse section (Fig. 150) has a slope rising upwards from the inner to the outer bank. Hence the free level, for any particle of the water near the outer bank, is higher than the free level for any particle in the same transverse section near the inner bank, but the tendency to flow from the higher to the lower level is counteracted by centrifugal action. Now the water immediately in contact with the bottom and sides of the course is retarded, and its centrifugal force is not sufficient RIMR-BENDS. 271 to balance the pressure due to the greater depth at the outside ofihe bend. This water therefore tends to flow from the outer bank towards the inner (Fig. 151), carrying with it detritus FIG. 151. which 'is deposited at the inner bank. Simultaneously with the flow of water inwards, the mass of the water must neces- sarily flow outwards to take its place. 13. Flow of Water in Open Channels of Varying Cross- section and Slope. Assumptions. (a) That the motion is steady. Thus the mean velocity is constant for any given cross- section, but varies gradually from section to section. (b) That the change of cross-section is also gradual. (c) That, as in cases of uniform motion, the work absorbed by the frictional resistance of the channel bed and sides is the only internal work which need be taken into consideration. Let Fig. 152 represent a longitudinal section of the stream. The fluid molecules which are found in any plane section ab at the commencement of an interval will be found in a curved surface dc at the end of the interval, on account of the different velocities of the fluid filaments. 272 CHANNELS OF VARYING CROSS-SECTION. Suppose that the mass of water bounded by the two trans- verse sections ab, ef comes into the position cdhg in a unit of time. Then the change of kinetic energy in this mass is equal to the algebraic sum of the work done by gravity, of the work / 9 done by pressure, and of the work done against the frictional resistance. Change of. Kinetic Energy. This is evidently the difference between the kinetic energies of the masses efgh and abed, since, as the motion is steady, the kinetic energy of the mass between cd and ^remains constant. Let A l be the area of the cross-section ab. " # t <( " mean velocity across this section. " v " " velocity at this section of any given fluid filament of sectional area a. Let v=u l V. Then A fa = 2(av) and = o. The kinetic energy of the mass abed = -2\a(u* 3*1** since = o and V CHANNELS OF VARYING CROSS-SECTION. 273 Now 2?^ -|- v is evidently positive. Hence the kinetic energy of the mass abed w > A.u* 2g l l = a AM ct being" a coefficient of correction whose value depends upon the law of the distribution of the velocity throughout the section ab. It is positive and greater than unity. Assume that a has the same value for the sections ab and ef. Then if A.,, ?/ 2 are the area and mean velocity at the transverse section eft the kinetic energy of the mass efgh . 2g * * Hence the change of kinetic energy in the mass under consideration wQ u? - u* Oi - , g 2 since ^2 = Q = A \ u r Work done by Gravity. Consider any fluid filament inn, the depth of m below the surface being y^ , and of n, y Let z be the fall in the surface-level from a to e. Then the fall from m to n and the work done by gravity on the elementary volume dQ in a unit of time 274 CHANNELS OF VARYING CROSS-SECTION. Work done by Pressure. The pressure per unit of area at m = wy^ -|~ / ; " " ," " M n = wy 2 + Pv p Q being the atmospheric pressure. Hence the work due to these pressures per unit of time Thus the total work done by gravity and by pressure = 2(w . dQ . z) = wQs for the mass under consideration. Work absorbed by Friction. Consider a thin lamina of water of thickness ds, bounded by the transverse planes xx, yy, the distance of xx from ab being s. Since the change of velocity is gradual, the mean velocity from xx to yy may be assumed to be constant. Let u be this mean velocity. " P be the wetted perimeter at the section xx. 1 * A be the area of the waterway at the section xx. Then the work absorbed by friction per second from xx to> yy P.ds.u. F(u), and the total work absorbed between ab and ef = <2 / being the distance between ab and ef. Hence CHANNELS &F VARYING CROSS-SECTION. 275 and therefore , = *'-*' . (* P w ds Take =/ and = m . Then *_ M 2 /'/ 2 =a^ -- 1_, I / -- ^y ^ m ^ 2 & Jo Wig (I) If the two planes ab and cf are indefinitely near one another (Fig. 153), the last equation evidently gives dz = u . du -\ ds, . - (2) g m 2g which is the fundamental differential equation of steady varied motion, dz being the fall of surface level in ,_ a the distance ds. In the figure aa is drawn parallel to the /^ bed and aa" is horizontal. The distance ^x a ' e may, without sensible error, be assumed equal to dz. FIG. 153. Also, a" a' = i . aa = i . ds, very nearly. Hence ids = a' a" --= a' e -f a"e = dh + dz, . . . (3) Substituting the value of dz from this equation in equa- tion (2), / . ds dh = u . du + . ds. , (A\ g m 2g Also, since Au = Q, a constant, A , du + u . dA o, and dA = x . dh. very nearly, if x is the width of the stream. 276 CHANNELS OF YARYIHG CROSS-SECTION. Therefore Adu + u* dh o, and hence, by equation (4), i.ds- dh = - *- .dh + u ~ g A ' m 2g Therefore i dh _ m 2g . m 2gi ds~~ ~^~x : ~^' ..- (5) I a -- ~. I a -. gA gA Let the position of any point a in the surface be defined by its perpendicular distance h from the bed and by the distance j of the transverse section at a from an origin in the bed. Then -r is the tangent of the angle which the tangent to the surface at a makes with the bed. It is positive or negative according as the depth increases or diminishes in the direction of flow, thus defining two states of steady varied motion. Between these there is an intermediate state defined by dh f w 2 - = o = i - , ds m 2g f " 2 and i = ~ - is the equation for steady flow with uniform m2g motion. Let U, M, H be the corresponding values of u, m, h in the case of uniform motion. Then ~ = b-M , ..... (6) M 2g M' .and eq. (5) becomes dh ds zr x u 1 x I a I a -: gA g A (7) CHANNELS OF VARYING CROSS-SECTION. 277 If the section of the channel is a rectangle, xh 'A xh, xhu = xHU, m = X -\-.2k Substituting these values in eq. (7), dh 7, and M = xH i Three cases will be considered and, in each case, a line PQ, drawn parallel to the bed, represents the surface of -uniform motion, H being the distance between PQ and the bed. CASE I. an* < gh and H < //, Fig. 154. -j- is positive, and therefore h increases in the direction of flow. Thus the actual surface MN of the stream is wholly above the line PQ. FIG. 154. Proceeding up-stream, h becomes more and -more nearly equal to //, so that the numerator of eq. (8), and therefore also dh j , approximates more and more closely to zero. clS Again, proceeding down-stream, h increases and u diminishes, so that both the numerator and denominator in eq. (8) approximate more and more closely to the value unity* dh and therefore , becomes more and more nearly equal to i* d-S the slope corresponding to uniform motion. 278 CHANNELS OF VARYING CROSS-SECTION. Hence up-stream MN is asymptotic to PQ, and down- stream MN is asymptotic to a horizontal line. This form of surface is produced when a weir is built across a channel in which the water had previously flowed with a uniform motion. CASE II. au* h, Fig. 155. -r is now negative, and the depth diminishes in the direc- tion of flow. Up-stream // increases and approaches H in value, so that MN is asymptotic to PQ, Down-stream h diminishes, n increases, and therefore the au 2 value of . is more and more nearly equal to unity. Thus, in the limit, the denominator in eq. (8) becomes zero, and therefore , = .00 . Hence theory indicates that at ds a certain point down-stream the surface line MN takes a direc- tion which is at right angles to the general direction of flow, This is contrary to the fundamental hypothesis that the fluid filaments flow in sensibly parallel lines. In fact, before the FIG. 155. limit could be reached this hypothesis would cease to be even approximately true, and the general equation would cease to be applicable. This form of water-surface is produced when there is an abrupt depression in the bed of the stream. Fig. 156 shows one of the abrupt falls in the Ganges canal as at first constructed. The surface of the water flowing freely over the crest of the fall took a form similar to MN below the line PQ of uniform motion. The diminution of depth in the CHANNELS OF VARYING CROSS-SECTION. 279 approach to the fall caused an increase in the velocity of flow, with the result that for several miles above the fall a serious erosion of the bed and sides took place. In order to remedy this, temporary weirs were constructed so as to raise the level FIG. 156. of the water until the surface line assumed a form MN' corre- sponding approximately to PQ. In some cases the water was raised above its normal height and a backwater produced. CASE III. au* > gh and H < h, Fig. 157. is negative and the surface line of the stream is wholly -above PQ. FIG. 157- dh If // gradually increases, u diminishes and ^- approximates to i in value. If h gradually diminishes, it approximates to H in value, dh and in the limit -. = o. ds CHANNELS OF VARYING CROSS-SECTION. Between these two extremes there is a value of h for which the denominator of eq. (8) becomes nil, viz., h = a , g and the corresponding value of ~j- is infinity. Thus one part of the surface line is asymptotic to PQ, the line of uniform motion, another part is asymptotic to a hori- zontal line, while at a certain point at which the depth is h = a , g the surface of the stream is normal to the bed. This is contrary to the fundamental hypothesis that the fluid filaments flow in sensibly parallel lines, and the general equation no longer represents the true condition of flow. In cases such as this there has been an abrupt rise of the surface of the stream, and what is called a " standing wave " has been produced. In a stream of depth H flowing with a uniform velocity [7, which IgH ich is > A / , V a to h v which is > construct a weir so as to increase the depth d FIG. 158. Then in one portion of the stream near the weir the depth aU* is > - , while further up the stream the depth is < STANDING WAVE. 281 U* Thus at some intermediate point the depth = a , the corre- '- sponding value of -y- being oo , and at this point a standing wave is produced. Now = Mi = Hi, U 2 and since H < ot , < a and therefore 2a which condition must be fulfilled for a standing wave. Bazin gives the following table of values of/": Nature of Bed. Slope |A = /) below which stand- ing wave is im- possible, In Metres per Metre. Standing Wave Produced. Slope in Metres per Metre (or Feet per Foot). Least Depth, in Metres. Very smooth cemented surface. . . . .00147 .00186 .00235 .00275 ( .002 -j .003 ( .004 ( .003 i .004 ( .006 {.004 .006 .010 ( .006 -< .010 ( .015 .08 03 .02 .12 .06 .03 36 .16 .08 I. O6 47 .28 Earth , A standing wave rarely occurs in channels with earthen beds, as their slope is almost always less than the limit, .00275. 282 STANDING WAVE. The formation of a standing wave was first observed, by Bidone in a small masonry canal of rectangular section. The width of the canal = o m .325 = x\ slope (= j of the canal = .023; " uniform velocity of flow = i m .69 = U\ ft depth corresponding to U = o m .o64 = H. A weir built across the canal increased the depth of the water near the weir to o m .287 = h r It was found that the * * uniform regime ' ' was maintained up to a point within 4 m . 5 of the weir. At this point the depth suddenly increased from O m .o64 to about o m . 170, and between the point and the weir the surface of the stream was slightly convex in form (Fig. 158). With the preceding data and taking a = I . I , and is therefore > I at a section ab, Fig. 159. At the section cd, .69 = 0. 377, and = -O55 therefore < Thus the expression I -- is negative for a section ab and positive for a section cd, so that / must change sign between dh these sections, and - will then ds become infinite. Consider a portion of a stream bounded by two transverse sections, ab, cd y in which a stand- ing wavs occurs, Fig. 159. Assume that the fluid filaments flow across the sections in -sensibly parallel lines. FIG. 159. STANDING WAVE. 283 Let the velocities and area at section ab be distinguished ^y the suffix i, and those at cd by the suffix 2. Then Change of momentum in di- ) fa I impulse in same direction, rection of flow ) Hence and therefore l -(2av*-2av*) = A l y l -A. i ? 2 , ... (9) <*> y l , j/ 2 being the depths below the surface of the centres of gravity of the sections ab, cd, respectively. Now, ^ = u^ + V r Therefore Also, as already shown, arid, neglecting V l as compared with aA v u* = A^i* -\- 3 Thus and hence a + 2 where a' = ~ , and is 1.033 if = i i . 284 STANDING WAVE. Similarly it may be shown that Thus equation (9) becomes '-A l u*)=Aj l -Aj r . . . (ID) Let the section of the canal be a rectangle of depth H l at ab and H 2 at cd. Then TT TT ti l H l = u 2 H 2 - -f = y^ ~ = y r Therefore, by equation (10), which reduces to H 2 = H l satisfies the equation and corresponds to a condition of uniform motion. Also, a'u? *_H^ + H, -- -~- In Bidone's canal, t = i m .6g, H^ = o m .o64. Substituting these values in equation (i i), the value of H^ is found to be O m . 16, which agrees somewhat closely with the actual meas- urements. N.B. The coefficients a and a' have not been very accurately determined, but their exact values are not of great importance. They are often taken equal to unity. RUHLMANN'S LAW. 285 14. Longitudinal Profile and Riihlmann's Law. In the preceding article, put F\i bj = i a -j in eq. (7), then = F. dh. If the transverse profile has been determined, the value of F corresponding to the depth h at any point O can be at once found and, by means of the last equation, the surface profile between the depths h and H can be easily plotted. Let F lt F 2 , F 3 , . . . be the values of F at a series of points at which the depths, differing successively by a small quantity dh, are h v , // 2 , ^ 3 , . . . respectively. Then s Y F l . dh\ ds 2 = F\ . dh ; F . dh; and the corresponding distances s\ , s. 2 , s^, . . '. of these points from O are ds.+ds, **+<*** . , ds.-l-ds, o i , . * i ~i , > > * .1 ~T~ i ^ 2 2 2 EXAMPLE. A cut of rectangular section, with a fall of i in 10,000, is 10 ft. wide and delivers 40 cu. ft. of water per second. At a certain point the depth is increased to 4 ft. by a dam. Assuming that the faces of the cut are not very smooth and that, consequently, .0001 maybe taken as an approximate value of b, then the depth, H, for uniform motion is given by = *y - loH 10+ 2 or 80 and an approximate solution of this equation is ff = 2.9 ft. The following Table can now be easily prepared for a series of depths, commencing at the dam and diminishing successively by 3 ins., a being unity: 286 RUHLM ANN'S LAW. k A P nt U I a * F ds * t>i 4.00 4 18 af T .991406 .000045 22031.2 55o8 3-75 37-5 17-5 2f | 991943 .000041 24193.7 6048 5,77& 3-5 35 17 2^y ! .992480 .0000372 26679.6 6670 12,137 32-5 16.5 i| .993017 .0000335 29642.3 7410 3.00 30 10 If 8 993554 .00003 8546 27,155 The tenth column gives the distances from the dam of the sections in which the depths are 3.75, 3.50, 3.25, and 3 ft. Riihlmann's formula for the distance between two sections, between which the depth of the water gradually increases from y + tfto Y+tfis =?{/-/)! the function frr being given by the following table: y H >(*) y H ^) y H 41) O.OI O.OO67 o-3 .3428 4 2.7264 O.O2 0.2444 0.4 .5119 5 2.8337 0.03 0.3863 0.5 .6611 .6 2.9401 O.O4 0-4889 0.6 v .7980 7 3-0458 0.05 0.5701 0.7 .9266 .8 3.1508 O.O6 0.6376 0.8 2.0495 9 3-2553 O.O7 0.6958 0.9 2.1683 2.O 3.3594 0.08 0,7472 I.O 2.2839 2-5 3-8745 O.O9 0-7933 i .1 2.3971 3-o 4-3843 0.10 0.8353 1.2 2.5083 3-5 4.8910 0.20 I. 1361 i-3 2.6179 4.0 5.3958 Applying this formula to the preceding example, in order to determine the distance between the 3- and 4-ft. depths f at the dam i.i = -3793, CHANNEL OF RECTANGULAR SECTION AND SMALL SLOPE. 287 and, by interpolation, = 1.4769. At the 3-ft. depth 7f = 9 = - 3448 ' and /() = .4323. Hence 2.9 .0001 (1.4769 .4323) = 30,293 feet. 15. Channel of Rectangular Section with a nearly Hori- zontal Bed. In this case i is very small and may be disre- garded in eq. (4), Art. 13, which may therefore be written in the form 2g m .. 2gmdu as = -p- ~^ah a.- -. / 2 / g u But xhu = Q = a constant, and therefore h . du -f- u . dh = o. Also, xh Hence x* h* . dh ax dh ~ ~ i. /T.) 7~H i -, /.N ~" ~ i i " Integrating, 'is' -~ log, (X + 2k) c being a constant of integration. 288 CHANNEL OF GREAT WIDTH AS COMPARED WITH DEPTH. Hence the distance s l s 2 between two points at which the depths are h^ and k. 2 (< //J is given by X* C^ * I *^ *> ~~r I^d / X -\-2ll The last term is usually very small and may be disregarded without appreciable error, and therefore ^ r** h\ih a formula by means of which the discharge may be found. 16. Channel of Great Width as compared with the Depth. In this case A = xh and P x, approximately. Therefore ;;/ h and MH. Also, Hence, eq. (7), Art. 13, may be written in the form ai \ a l "ft ~ l ~ 3 " ds \H dh Take s = JT~~ fj> ? being the rise or fall above or below the surface of uniform motion. Then dh = H . dz, and ai ids ~ _ _ Hdz~ ""^ 8 -l*. CHANNEL OF GREAT WIDTH AS COMPARED WITH DEPTH. 289 Integrating, being a constant of integration. This equation may be written 2s Tl 7 / and between any two points and -, , . (3) j/ change in depth argument being = .- 7/z the case of a dam built across a channel in which the water had previously flowed with a uniform motion, Case I, Art. 13, in the limit, si = h zH = oo , and therefore, by eqs. I and 2, <(*) + <: == O = g log, I + i 7-- tan " I + r= -^-f + ' and <: .9069. The following Table, calculated by Tutton, gives the value of the backwater function, (z), in the case of a dam : 290 BACKWATER FUNCTION. y H *(*) y H *(*) y H *(*) y H *(*) o.ooo 00 .072 .7812 .285 .3860 .68 1945 .001 2.1837 .074 .7727 .290 .3816 .69 .1918 .002 .9530 .076 .7644 .295 3773 .70 .1892 .003 .818.1 .078 7564 .300 3730 7i .1867 .004 .7225 .080 .7486 .305 .3689 .72 .1843 -005 .6485 .082 .7410 310 .3649 73 .1819 .006 .5881 .084 .7336 315 .3609 74 1795 .007 5379 .086 .7264 '.320 3570 75 .1772 .008 .4928 .088 .7194 .325 3532 .76 .1749 .009 4539 .090 7125 330 3495 77 .1727 -OIO .4191 .092 .7058 335 .3458 .78 1705 .Oil .3877 .094 6993 340 .3422 79 .1684 .012 .3586 .096 .6929 345 3387 .80 .1663 .013 .3327 .098 .6866 350 3352 .81 .1642 .014 .3082 . IOO .6805 355 .33i8 .82 .1622 .015 -2855 .105 .6658 .360 3285 83 . 1602 -Ol6 .2644 . no .6518 .365 3252 .84 1583 .017 .2446 IIS .6387 370 .3220 85 .1564 .018 .2258 . 1 20 .6260 375 .3189 .86 .1546 .Dig .2081 .125 .6139 .380 3158 87 1528 .020 1913 .130 .6024 .385 .3127 .88 1510 .021 1754 135 .5913 390 3097 .89 .1492 .022 .1602 .140 .5807 395 .3068 .90 1475 .023 .1457 145 .5706 .400 .3039 .91 .1458 .024 1319 .150 .5608 .41 ' .2982 .92 .1441 .025 .1186 155 5514 .42 .2928 93 .1425 .026 .1059 .160 .5423 43 2875 94 .1409 .027 .0936 .165 5335 44 .2824 95 1393 .028 .0817 .170 5251 45 .2774 .96 1377 .029 .0704 175 .5169 .46 .2726 .97 .1362 .030 .0595 .180 .5090 47 .2680 .98 1347 .032 .0387 .185 .5014 .48 .2634 99 1332 -034 .0191 .190 -4939 ^49 .2590 1. 00 .1318 .036 .0007 195 .4867 50 .2548 1.05 .1250 .038 .9833 .200 .4798 5i .2506 I. 10 .1187 .040 .9669 .205 4730 52. .2465 15 .1128 .042 9512 .210 .4664 53 .2426 .20 .1074 .044 .9364 .215 .4600 54 .2388 25 .1024 .046 .9223 .220 .4538 55 2351 30 .0979 .048 .9087 -225 4478 56 .2314 35 .0936 .050 8957 .230 .4419 57 .22-79 .40 .0894 .052 8833 235 4363 58 .2245 45 .0856 054 .8714 .240 .4306 59 .2212 50 .0821 .056 .8599 -245 4251 .60 .2179 55 .0788 058 .8488 .250 .4198 .61 .2147 .60 .0758 .060 .8382 255 .4145 .62 .2Il6 65 .0728 .062 .8279 .260 .4096 63 .2086 .70 .0700 .064 .8179 265 .4046 .64 .2056 75 .0674 .066 .8083 .270 .3998 65 .2O27 i. 80 .0650 .068 .7990 275 3951 .66 .1999 1.85 .0626 ,O7O .7899 .280 3905 .67 .1972 1.90 .0604 BACKWATER FUNCTION. 291 y H* *(*) y H *(*) y H *(*) y H <*>(*) i-95 .0584 3-4 .0260 7.0 .0078 20. o .0011 2.OO 0564 3-5 .0248 8.0 .0062 25.0 .0007 2. I .0527 3 6 .0237 9.0 .0050 30 o .0005 2.2 .0494 3-8 .0218 IO.O .0041 35-o .0004 2-3 . 0464 4.0 .0201 II. .0035 40.0 .0003 2.4 0437 4.2 .01.85 12.0 .0030 45-o .0002 2-5 .0412 4-4 .OI72 13-0 .0026 50.0 .0002 2.6 .0389 4.6 .Ol6o 14.0 .0022 99.0 .0001 2.7 .0368 4-8 .0149 15-0 .0019 IOO.O .0001 2.3 0349 5-o .0139 16.0 .0017 oo .0000 2.9 0331 5-5 .0118 17.0 0016 3-o .0314 6.0 .OIOT 18.0 .0014 3- 2 .0285 6-5 .0089 19.0 .0013 NOTE. The corresponding Table, deduced by Bresse, whose argument H 4- y y is = I -f- i ma Y be at once obtained from the above by adding i H H to Tutton's argument. In the case of a fall, Case II, Art. 13, in the limit si = h = zH o, and therefore, by eqs. (i) and (2), .and I 7f The following Table, calculated by Tutton, gives the value of the backwater function, cf>(s\ in the case of a fall : 292 BACKWATER FUNCTION. V 77 *() y H #*) 1 y I H *(*) y H *(*) 0. 00 .040 .5448 .20 .9504 45 5753 .001 2.7876 .042 5279 .21 9303 .46 5633 .002 2.5562 .044 5TI7 .22 .9109 47 5515 .003 2.4207 .046 .4962 23 .8922 .48 .5398 .004 2.3244 .048 .4813 .24 .8741 49 5282 .005 2.2497 .050 .4670 25 .8566 50 5167 .006 2.1885 055 4335 .26 8395 5i 5054 .007 2.1368 .060 .4027 .27 .8229 52 .4941 .008 2.0920 .065 3743 .28 .8068 53 .4829 .009 2.0525 .070 3479 .29 .7910 54 4717 .010 2.0171 075 3231 30 7756 55 .4607 .012 9554 .080 .2999 31 .7606 56 4497 .014 .9036 .085 .2779 32 7458 57 .4388 Ol6 .8584 .090 2571 33 7313 58 .4279 .018 .8185 095 -2372 34 .7172 59 .4171 O2O .7827 . IOO .2185 35 7033 .60 .4064 .022 .7502 .11 .1831 36 .6896 65 353f> .024 .7206 .12 .1504 37 .6762 .70 .3019 .026 .6936 13 .1201 38 .6629 75 2510 .028 .6678 .14 .0918 39 .6499 .80 .2004 .030 .6441 15 .0651 .40 .6371 .90 . IOOI .032 .6219 .16 0399 .41 .6244 I.OO .0000- 034 .6010 -17 .Ol6o .42 .6119 .036 5813 .18 9931 43 5995 .038 .5626 .19 9713 j 44 5873 1 NOTE. Bresse uses the same value .9069 for the con- stant c both for a dam and for a fall. His argument in the latter case is H- y H V -~ x and to obtain Bresse 's Table ri from the above, the argument adopted by Tutton is subtracted from I, and .6046 from the value of (z). y Dupuit, again, uses the argument ~y, and his Tables may be obtained from those given by Tutton by equating his back- water function to y 1.4158 -f- (z) for a rise, ri and to 2.0204 ft- for a fall. CHANGE OF CROSS-SECTION. 293 , Dupuit neglects the term - - and includes in his back- , the surface of FIG. 1 60. steady motion being PQ above A A, and RS below CC. On approaching A A the surface gradually rises and reaches its greatest height QT ' = z above PQ at AA. This is followed by a gradual fall to the surface of steady motion RS at CC. Let k l , /2 2 (> 7/j) be the depths corresponding to steady motion above AA and below BB, respectively. " /#! , #/ 2 be the mean hydraulic depths above A A and below BBj respectively. and u. 2 be the mean velocities of flow above A A below BB, respectively. 2 9 4 CHANGE OF CROSS-SECTION. Then, disregarding the effect of surface resistance between A A and CC, or If the section is a rectangle, But Therefore and If the width is great as compared with the depth, > = ~ and ;;/ 2 = , approximately. 2 2 Therefore and CHANGE OF CROSS-SECTION. 295 CASE II. A channel of slope i, in which the flow is steady, PQRS being the surface of steady motion, gradually contracts from a width A A = B, to a narrower width at CC. The FIG. 161. channel remains narrow for a limited distance CD and then gradually enlarges to its original size at E. On approaching A A the surface rises, attains its greatest height QT above PQ at A, falls to V at C, then to a point W below PQRS at D, and finally suddenly rises from W to the surface of steady motion at R. Let z be the depression of W below PS. 4 ' B, B l be the widths at D and E. " u, u i be the mean velocities at D and E. Then z - where a may be taken = I . i . If the section is a rectangle, B(k l z)u B l u l h r Therefore, a cubic equation giving z. 296 CHANGE OF CROSS-SECTION. The surface DE may now be plotted, and QT may be found as in Case I. These expressions also give, approximately, the depression below the surface PQRS of steady motion when the channel has its section suddenly changed by such obstructions as bridge-piers. FIG. 162. On approaching the pier ends the water-surface gradually rises to the maximum height T above PS, then falls to XV below PS between the piers, and finally rises again to the sur- face of steady motion on passing into the open channel. Let B^ , B be the distances between the axes and the inner faces of the piers. Let H be the depth below XY. Let z be the fall from T to X. Then, according to Bresse, the value of z is given by the empirical formula B?(H 4- c c being a coefficient of contraction and having an average value of about .8. Also, Q is the discharge for the width B l of the channel. * This formula, although generally adopted, is open to question. Bresse considers that an equally correct approximation is obtained at a distance / /? of 2O(Bi B) from the contraction by taking z 2oiB [ - V B GAUGING. 297 18. Gauging of Streams and Watercourses. The amount of* flow Q in cubic feet per second across a transverse section of A sq. ft. in area is given by the expression u being the mean velocity of flow in the section in feet per second. If the longitudinal profile and several transverse sections of a channel can be plotted, the volume of flow may be calculated by means of eq. (i), p. 275. Let ?/ 1 , u 2 , . . . u n be the mean velocities, A l , A 2 , . . . A n the areas, and P l , P 2 , . . . P H the wetted perimeters of n sec- tions of the channel at the specified distances ^ , / 2 , . . . / w _ t apart. Then z, the fall in the free-surface level, which may be found from the longitudinal profile, is given by the equation z = a'"---- 1 - + I ~ds, in which / / / -u/ f - mi h T iH- 2-1 -i - . w a - -^' and a may be taken = I . i . But Aji^ Au = Q /. u - - A n 2t n , and m p- Therefore and Q can be calculated as soon as the integration has been effected, which may be possible if P and A are known functions of s. An approximate value of the integral may be found graphically as follows : P Plot, as ordinates, the values of -, at the n sections, and yi join the upper ends of those ordinates. The area between the 298 GAUGING. extreme ordinates, the axis, and the line thus determined is the value required. FIG. 163. Generally speaking, however, the above method of gauging the flow of a stream is not very accurate, on account of the errors in the values of P, A, and the integral. More correct results are obtained by determining the mean velocity. 19. Determination of the Mean Velocity u. METHOD I. The most convenient method for gauging small streams, canals, etc., is by means of a temporarily constructed weir, which usually takes the form of a rectangular notch. The greatest care should be exercised to insure that the crest of the weir is truly level and properly formed, and that the sides are truly vertical. The difference of level between the crest of the weir and the surface of the water at a point where it has not begun to slope down towards the weir is best estimated by means of Boyden's hook-gauge, Fig. 164. This gauge consists of a carefully graduated rod, or of a rod with a scale attached, having at the lower end a hook with a thin flat body and a fine point. The rod slides in vertical supports, and a slow vertical movement is given by means of a screw of fine pitch. A stiff vertical rod, with a sharp point, having been placed at 5 to 8 ft. from the back of the weir, with the point on a level with the weir crest, water is run into the flume until it rises slightly above the crest, producing a capillary elevation at the point. The FIG. 164. water is then allowed to subside until this elevation is barely" GAUGING. 299 perceptible, when the rod may be removed. A hook-gauge is hext placed in the same position, and the hook is slowly raised until a capillary elevation is produced over its point. The hook is then slowly lowered until the elevation becomes almost imperceptible, when a reading is taken corresponding to the level of the crest of the weir. More water now flows into the flume and over the weir. As soon as the motion has become steady, the hook is raised and the point adjusted at the surface in the manner just described. A second reading is taken and the difference between the two readings is the head of water over the crest. In ordinary light, differences of level as small as the one- thousandth of a foot can be easily detected by the hook- gauge, while with a favorable light it is said that an experi- enced observer can detect a difference of two ten-thousandths of a foot. Such differences, however, cannot be measured under the ordinary conditions of practical work. METHOD II. A portion of the stream which is tolerably straight and of approximately uniform section is defined by two transverse lines O^AB, OfD at any distance 5 ft. apart. FIG. 165. The base-line O^O 2 is parallel to the thread EF of the stream, and observers with chronometers and theodolites (or 300 GAUGING. sextants) are stationed at O l , O v The time T and path EF taken by a float between AB and CD can now be determined. At the moment the float leaves AB the observer at O l signals the observer at O 2 , who measures the angle O^Of.^ and each marks the time. On reaching CD the observer at (9., signals the observer at O l , who measures the angle O^O^F, and each again marks the time. Experience alone can guide the observer in fixing the dis- tance 5 between the points of observation. It should be remembered that although the errors of time observations are diminished by increasing 5, the errors due to a deviation from lines parallel to the thread of the stream are increased. A number of floats may be sent along the same path, and their velocities \-j?j are often found to vary as much as 20 per cent and even more. Having thus found the velocities along any required number cf paths in the width of the stream, the mean velocity for the whole width can be at once determined. Surf ace -floats are small pieces of wood, cork, or balls of wax, hollow metal and wood, colored so as to be clearly seen, and ballasted so as to float nearly flush with the water-surface and to be little affected by the wind. Sub surf ace -floats. A subsurface float consists of a heavy float with a comparatively large intercepting area, maintained FIG. 166 FIG. 167. at any required depth by means of a very fine and nearly GAUGING. vertical cord attached to a suitable surface-float of minimum immersion and resistance. Fig. 166 shows such a combina- tion, the lower float consisting of two pieces of galvanized iron soldered together at right angles, the upper float being merely a wooden ball. Another combination of a hollow metal ball with a piece of cork is shown by Fig. 167. The motion of the combination is sensibly the same as that of the submerged float, and gives the velocity at the depth to which the heavy float is submerged. Twin-floats. Two equal and similar floats (Fig. 1 68), one denser and the other less dense than water, are connected by a fine cord. The velocity (T^) of the combination is approximately the mean of rr the surface velocity (v s ) and of the velocity (v d ) at the depth to which the heavier float is sub- Thus merged. and therefore v f = FIG. 168. so that v d can be determined as soon as the value of v t has been observed and the value of v s found by surface-floats. Velocity-rod. This is a hollow cylindrical rod of adjustable length and ballasted so as to float nearly vertical. It sinks almost to the bed of the stream, and its velocity (?',) is ap- proximately the mean velocity for the whole depth. Francis gives the following empirical formula connecting the mean velocity (v m ) with the observed velocity (v r ) of the rod : FIG. 169. OI2 . I J6. V w 3 02 GAUGING. d being depth of stream, and d' the depth of water below bottom of rod ; but d' should not exceed about one fourth of d. METHOD III. Pitot Tube and Darcy Gauge. A Pitot tube (Figs. 170 to 172) in its simplest form is a glass tube with a right-angled bend. When the tube is plunged vertically into the stream to any required depth z below the free surface, with its mouth pointing up-stream and normal to the direction of flow, FIG. 170. FIG. 171. FIG. 172 the water rises in the tube to a height h above the outside surface, and the weight of the column of water, .7 -+- // high, is balanced by the impact of the stream on the mouth. Hence (Chap. V) wA( wAz kwA , and therefore A being the sectional area of the tube, 21 the velocity of flow at the given depth, and k a coefficient to be determined by experiment. A mean value of k is 1.19. With a funnel-mouth or a bell-mouth Pitot found k to be 1.5. This form of mouth, however, interferes with the stream-lines, and the velocity in front of the mouth is probably a little different from that in the unobstructed stream. The advantages of tubes of small section are that the dis- turbance of the stream-lines is diminished and the oscillations GAUGING. 303 of the column of water are checked. Darcy found by careful measurement that the difference of level be- tween the surfaces of the water-column in a tube of small section placed as in Fig. 170, and of the water- column placed as in Fig. i 71 with its mouth parallel to the direction of flow, is almost exactly equal to When the tube is placed as in Fig. 172 with its mouth pointing down- stream and normal to the direction of flow, the level of the surface of the water in the tube is at a depth h' below the outside sur- face, and *' = *' ft/ - fa . 2^ where k' is a coefficient to be determined by ex- periment and a little less than unity. In this case the tube again obstructs the stream- lines. Pitot's tube does not give measurable indi- cations of very low veloci- ' ties. A serious objection FIG. 173. 34 GAUGING. to the simple Pitot tube is the difficulty of obtaining accurate readings near the surface of the stream. This objection is removed in the case of Darcy's gauge, shown in the accom- panying sketch, Fig. 173. A and B are the water-inlets ; C and D are two double tubes; E is a brass tube containing two glass pipes which communicate at the bottom with the water-inlets and at the top with each other, and with a pump F by which the air can be drawn out of the glass pipes, thus allowing the water to rise in them to any convenient height. Thus Darcy's gauge really consists of two Pitot tubes con- nected by a bent tube at the top and having their mouths at right angles or pointing in opposite directions. If Ji is the difference of level between the water-surfaces in the tubes when the mouths are at right angles," then and Darcy's experiments indicate that k does not sensibly differ from unity. When the mouths point in opposite directions, let /?, , /i 2 be the differences of level between the stream-surface and the surfaces of the water in the tube pointing up-stream and the tube pointing down-stream, respectively. Then and therefore h^ -\- h 2 (k l -{- -i- K- , where k = k^ -)- k 2 . GAUGING. 30$ k having been determined experimentally once for all, the difference of level (= h l -\- // 2 ) between the columns for any given case can be measured on the gauge and the value of u can then be found. A cock may be inserted in the bend connecting the two tubes, and through this cock air may be exhausted and a par- tial vacuum created in the upper portion of the gauge. The water-columns will thus rise to higher levels, but the difference between them will remain constant. Thus the surface of the column in the down-stream tube may be brought above the level of the outside surface, and the reading is then easily made. Sometimes the gauge is furnished with cocks at the lower parts of the tubes, and if these cocks are closed when the measurement is to be made, the gauge may be removed from the stream for the readings to be taken. METHOD IV. Current-meters. The velocity of flow in large streams and rivers is most conveniently and most accurately ascertained by means of the current-meter. The earliest form of meter, the Woltmann mill, is merely a water- mill with flat vanes, similar in theory and action to the wind- mill. When the Woltmann is plunged into a current, a counter registers the number of revolutions made in a given interval of time, and the corresponding velocity can then be determined. This form of meter has gone out of use and has been replaced by a variety of meters of greater accuracy, of finer construction, and much better suited to the work In its simplest form the present meter consists of a screw-propeller wheel (Fig. 1/4), or a wheel with three or more vanes mounted on a spindle and connected by a screw-gearing with a courier which registers the number of revolutions. The met^r is put in or out of gear by means of a string or wire When a cur- rent velocity at any given point is to be found, the reading of the counter is noted, the meter is sunk to the required position, and is then set and kept in gear for any specified interval 0f 306 GAUGING. time. At the end of the interval the meter is put out of gear and is raised to the surface, when the reading of the counter is again noted. The difference between the readings gives the number of revolutions made during the interval, and the velocity is given by an empirical formula connecting the velocity and the number of revolutions in a unit of time. The vane V is introduced to compel the meter to take a direction perpendicular to that of the stream-lines, but this may not necessarily be perpendicular to the axis of the stream. The slight error due to this discrepancy is usually disregarded in practice. In order to prevent the mechanism of the meter from being FIG. 174. FIG. 175. injuriously affected by floating particles of detritus, Revy enclosed the counter in a brass box, Fig. 175, with a glass face, and filled the box with pure water so as to insure a con- stant coefficient of friction for the parts which rub against each other. In the best meters, however, the record of the number of revolutions is kept by means of an electric circuit, Fig. 176, GAUGING. 307 which is made and broken once, or more frequently, each resolution, and which actuates the recording apparatus. The time at which an experiment begins and ends is noted, and the revolutions made in the interval are read on the counter, which may be kept in a boat or on the shore, as the circumstances of the case may require. The meter is usually attached to a FIG. 176. FIG. 177. suitably graduated pole, so that the depth of the meter below the water-surface can be directly read. In deep and rapid water the meter must be held by a wire cord which will usually require to be guyed to a forward line. The mean velocity for the whole depth at any point of a stream may be found by moving the meter vertically down and then up, at a uniform rate. The mean of the readings at the two surface" positions and at the bottom position will be the number of revolutions corresponding to the mean velocity required. The* mean velocity for the whole cross-section may also be deter- mined by moving the meter uniformly over all parts of the section. The meter should be rated both before and after it is used. This is done by driving the meter at different uniform speeds 308 GAUGING. through still water. Experiment shows that the velocity if and the number of revolutions n are approximately connected by the formula v = an -f- b, where a and b are coefficients to be determined by the method of least squares or otherwise. Exner gives the formula V Q being the velocity at which the meter just ceases to revolve. OTHER METHODS.- Many other pieces of apparatus for the measurement of current velocities have been designed. Perrodil's hydrodynamometer, for example, gives the velocity directly in terms of the angle through which a vertical torsion-rod is twisted, and in this respect is superior to the current-meter. The tachometer or hydrometric pendulum (Fig. 178), again, connects the velocity with the angular deviation from the vertical of a heavy ball suspended by a string in the current. IX - N ~- v Hydrometric and torsion balances have also been devised, but thev must be regarded rather FIG. 178. as curiosities than as being of any real practical use. Having found the maximum surface velocity, v s , at any point in a watercourse, by one of the above methods, then (Art. 10, p. 259) the mean velocity of the whole section is given by the empirical relation If the transverse section of the waterway, at the point in ques- GAUGING, 309 tion, is plotted and its area, A, measured, the discharge, Q^ may be at once calculated by means of the formula * <2 = \Av s . Again, selecting an approximately straight length of channel, let x be the distance from the origin of a particle in the surface filament of maximum velocity. Then the velocity of this particle is -, and therefore Hence Qt = - I Adx, SJo t being the time in which the float passes over the distance s. If this distance is now divided into n equal divisions, and if A^, A^ A 2 , A 3 , . . . A n are the areas of the waterway at the commencement, at the (// I ) intermediate division points and at the end of the length s, then, by Simpson's rule, The integration may also be at once effected if A is given as a function of x. Again, if H is the depth of steady motion, -- b ~ A* and if the width B of the channel is large as compared with H, approximately, and A = BH. m GAUGING. Therefore Q = At any given point in the stream B may be considered con- stant, i is also constant, and a coefficient m may be substituted for B\ . The actual depth //, which may be read on a fixed V b vertical scale at the point in question, differs from H by a certain quantity ;/. Thus the last equation may be written in the form Q = ;,/(// -f ;/)t, a convenient expression which is sometimes used to determine the volume of flow in wide rivers. The coefficients m and n are constant at the same point for all depths, but vary from point to point. TABLE GIVING THE VALUES OF m AND ;/, THE UNIT BEING A METRE OR A FOOT. t n i t Locality. Metres. Feet. Metres. Feet. Authority. Mantes bridge on the Seine QC *6* 7 2 "\ Roanne bridge on the 1 80 1070 .25 .82 Graeff Come bridge on the Adda 100-3.2 h. 594-5.8 h. .O .O Lombardini NOTE. From an examination of a large number of gaugings Bresse infers that u = .8527, gives better average results than u .Sv s (Art. 10 and p. 259). The latter, however, is equally safe unless it is necessary to pro- vide against floods. (Ann. des Fonts et ChaussJes, 1897.) GAUGING. 1000 M M H* d ^* O^ c*"i "^* oj u~> o^oo r^* rf* en w - X * o ojin-^u^r^oc^or^ococoocT^O^ ^ u->u->r^OTO*"^ < <1-r^Tj-i".or>.^TOi M o s O^ co r-*- r^- r->- o O O vo to u"> ir> ,-. H4 HH O O O O W S 000000 z - . . 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VJ ^ s H i H OH "J O ^ W P4 o w Mlf OQ tl ft i^r^ooco M M c c r-^.co o N tor^o cno oi M i-i o OOOQO r^o o >-i cnior^ O O O O O O O O co r^ M CM o O 't O O c^co i^- r->-o * M O O O O O ID r^ o** O c*J ^f *D t^co O w CM ^~ ID o VDVDIOIO\DOID1DXDIDIDIDIDIDIDDIDU11DIDIDDVDD M CO'I- & O-ICMCO >* \r> or^ co C* O M w i^-- ooooodcd- coco- cocol co- cd- (>- C>- O^- |S ikl M coo C^N vor^O 'frr^.O cor-O cot^ OO r^r^r-aoccco M -tr>O CM DOO M O coo O coo O* coo O co r->. O O O M M M M CM CM cococo w O O w c* co -t >DO r^co 3*4 8 8 st D ri MH y w S c/} ii o i rt c^ *> 7. s* EL, ii GAUGING. s co O co O ^f M~> O* >"* *t O co 8i_ M M i-H M N - W N C* 04 - COCO- COCO- COCO" TT OOOOOO OOOOO OO OO OO O 1. N O C> \f. o r* co O^ O "-i co ^f m C4 M o O^co r* o w^ ^ co C4 M o o^ co r^> o >^ ^t* co (NNat-HWMt-CMWWWM OOO. OO OO. cococococococococococ<-ico- cocococo- coco- coco- O ir> O O , C^ - - - - N ' *: i- ! ? O coo O coO O O ONO CTWO O>N mco 1-1 rfoo N r> \oo !* r c>oo oo.-oo 0^000 O O H -i GAUGING. ga K o CT> O^ CT^co oocooooooo vo o vo o O *O 10 o r^-oo 1-1 co to lO CO O N CO to O M en-3- co co co ro co w tooo to *- oo TfM r^Ti-i-i i^-t-o r^coo r^coOO coCT> <~< OJ M C^ co*^--tto>oO t^oooo CT>O O 1-1 CM W co-3--3-ioxo GAUGING. o " o" o"" o ' o"" O M OO N O ""> N oo *r>~cToo~ "^ t- co ini-ioo TWQO -fO or^t>.coOOMr-iMMcnco-tir)inor^r^coooi-i GAUGING. 317 : U O e>i ^ o r^ O t^ ^ "< oo en o ^o o o O en co N o O^ en t^* *H o Q as be o ; coooooooooooooooooooooooco OOOOOOOOOOO Tfoo CN Oenr^oioo r>-oo O \n O ^M Oooo l^r^oo Oen^t- O jj * l 1 5 t"?^ t r??'?'7""?"':T t :':T'r c ri t ? i ?T o - -g "i < u 03 6 Us OOO u->invn-^-encno CM H- >-> O O O Ooo oo oo co r^ t-^ r>. ^ M en^oo N rto OM en 1- Ooo OW ii . Vj Q O O rt i CNd CM tNwt-iOOOOO C^CO OOOO OO Hs (H >? t^ oo OOwcnrj-u->oooO OH Men *t \r> .s 11 2 V 01 "^ H *"* (J 01 u d M o O^ ^^ o ^> ^ en (MM O O oo r^. o d N ^ d. (NtNMCNMMdCN^ CMCN^ ^ Mo M d W a s ? I b O O -' o * O * O * ^ O ' O'"**'*O'"'* M en en O OOO H g a -. "* 8 < o 5 d en ^" u~j o t^ oo O PQ jj u s 4; r^ o u-> ^f O OOO ^ 8 888 fa r>> o M en O V) W D 1 i tu bjo I fc, , -1- Tf rt CO vQ > o .ti i W O M N en P o > D g-s Metre. c 00 00 00 00 1 8 a c? tn'en^-^^co'd Etc d ^^cc CT S>< ^-w M ^ J *oj he OOOOOOO r-r r^oo ooooooooOOMMMdN CQ *- 41 C 111 s O d rfo ooOOOOOOOOOOOOOOOOOOO OOOOOO M d en-fu->o r^oo oo >-< d cn^vno t^-co GAUGING. W NO M CO -t O^ tno w o X S *> s 52 "^~ ^ u^ to o vo r^* t^> oo oo O^O^OQvOO^OO^^O^ QO M C")i--OO O^OO M N o or^-o lOTftnc^ MP-.MQOOOOOO 1 2 4-> /, j O N OQ\M mr^O^M Tf- 1 3 S w s ^ oJwiNN'e^cnfScnTt-^- lommu-iioirjiniotoio Oco MOO "i- o r^Tj-M r^-OO ioir> l o-ti--rfco CM CO CO CO CO CO CO CO CO CO 't in r^ o^ -i co 10 ^ CO- CO CO CO ^ TJ-1 ^ ^ rt {/) to I > 4-> I vl r^ t M QO n c< r^ r^ r^ o o o 1^- J^I^ l^t~>-. !->. o- oo- oo- o- o v -C fc V> V u W s ^ O r^oo O^ O M w w - N N pi co co : co - o o o o o vo o W> Tf CO N M O vn in r> u-> m -> M- CMW-u C^O- W, ^ o rt s g 1 o- o'-- o'--- co co co oo m oo oo < -S * 1 ^ 1 1 8 & O O t-H f, S 4J s ^ t^ r^ r^ S N H.% . . , ^ u en to 1 S s Tf **" M M i 1! Si C/J 1 CO Tf > 4) K J3 w t5 > J s s ^ oo oo in 8 *.** O" N rtO CO O N ^O oo r~.i-i t^i^-mrt-oj OoovO 0^ bfi N coco^mo rococo O W^ |al V V OO>nOoOmOmO OO M mr^o c^ mt^O co ^~ T^- TJ- ^- i/-) \D tn in o GAUGING. VALUES OF b AND c, FOR THE SIX CLASSES I TO VI, P. 250, IN BAZIN'S NEW FORMULA, bv 1 = mi, OR v = cVini, WHERE = 87 i/^7/, OR = 157.6 t/,,/, ACCORDING AS THE ( 1 + -Z=?\ = \ Vmt UNIT IS A METRE OR A FOOT. Value of /, the Unit being a Metre Foot. CLASS I, the Unit being a CLASS II, the Unit being a CLASS III, the Unit being a Me b tre. c Fo b ot. c Me b tre. c Fo b ot. c Me b tre. c F b oot. c ' \ 5 .16 .0146 68.5 .00445 124.1 0197 50-7 .00600 91.8 0352 28.4 .01073 5 1 -* .06 .20 0143 69.8 .00436 126.4 .0190 52-6 .00579 95-3 Q33 1 30.2 .oioo9 54-5 .07 23 .0141 70.9 .00430 128.4 .0185 54-2 .00564 98.2 0315 3!-7 .0096 56.3- .08 .26 .0139 71.8 .00424 130.0 .0180 55-6 .00549 00.7 .0302 33-i .0092 59-9- .09 30 .0138 72-5 .00421 131-3 .0176 56.7 .00536 02.7 .0291 34-4 .oo887 62.3. . 10 33 0137 73- 1 .00418 132.4 0173 57-7 .00527 04-5 .0282 35 -S .oo8 5 6 64- a . ii 36 .0136 73.6 .00415 *33-3 .0170 58.7 .00518 06.7 .0274! 36.^ .00835 66.1 . 12 39 OI 35 74.1 .00411 124.2 .0168 59-5 .00512 08.3 .0268 37-4 -oo8i7| 67.7 J 3 43 0134 74.6 .00408 1 35-* .0166 60.2 . 00506 09.0 .0262 38-2 .oo 79 9 69.2 .14 .46 0133 75-o .00405 135-8 .0164 60.9 . 00500 10.3 .0256 3Q-o .0078 70.6 15 49 75-3 " T 3 6 -3 . o 1 63 61.5 .00497 11.4 .0252 39-7 .0076 71.9 .16 52 .0132 75-6 .00402 136.9 .0161 62.1 .00491 12-5 .0247 40.5 oo 75 3 73-4 - 1 ? 56 " 75-9 * k '37.4 .0160 62.7 .00488 13.6 .0243 41.2 .0074! 74-7 .18 59 ,OT 3 IJ 76.2 .00399 138.0 .0158 63.2 .00482 14.5 .0240 41. 8 .00732 75-8 .19 .62 " 1 76.5 138.5 -0157 63-6 .00479 15.2 .0236 42.4 .00719 76.9 .20 65 .0130 76.7 .00396 138.9 .0156 64.1 .00476 16.1 0233 42.9 .0071 77-7 .21 .69 " 76.9 " '39-3 0155 64-5 .00473 16.8 .0230 43-5 .0070' 78.8 . 22 72 " 77- 1 11 1 39-6 .0154 64 9 .00470 17.6 .0228 44-o .00695 79-7 23 75 .0129 77-3 .00393 140.0 .0153 65-2 .00467 18.1 .0225 44-4 .00686 80. i 2 4 79 77-5 140.4 65-5 18.6 .0223 44-8 .00680 81.2 -25 .82 " ! 77-6 " M-5 .0152 6 5-9 .00463 '9-3 .0221 45-3 . 00674 81.7 .26 85 " 77-8 11 140.9 .0151 66.2 .00460 19.9 .0219 45-7 .00668 82. a .27 .88 .0128! 78.0 .00390 I 4'-3 .0150 66.5 00457 20.4 .0217 46.1 .00662 83-5 .28 .92 " 78.1 J 4i 5 66.8 21 .O .0215 46.5 . 00654 84.2 .29 95 ;; 1 78.3 " 141.8 .0149 67.0 .004^54 21-4 .0213 46.9 .00649 84.9 .30 .98 78.4 44 142.0 67-3 21.9 .0211 47-3 .00643 85-7 3* .02 " 78.5 " 142.2 .0148 67.6 .00.451 22.5 .0210 47-6 .00639 86.2 S 2 05 .0127 78. b -00387 142.4 4 * 67.8 22-9 .0209 47 9 .00637 86.7 -33 .08 " 78-8 14 142.7 .0147 68.0 .00448 2 3 .2 .0208 48.2 .00634 87.3 34 . 2 789 " 142.9 68.2 * * 2 3.5 .0206 48.5 .00628 87 8 35 . 5 79.0 " 143 - 1 .0146 68.4 .00445 23-9 .0204 48.8 .00622 88.3. 36 . 8 " 79.1 " *43-3 44 68.6 24.2 .0203 49-2 .00619 80. r 37 . i .0126 79.2 .00384 M3-5 .0145 68.8 .00441 2 4 .6 .0202 49-5 .00616 89 6 .38 e l 4 * 69.0 44 25.0 .0201 49.8 .00613 90.2 39 . 8 " 79-3 U 143.6 .0144 69.2 .00438 25-4 .0200 50.1 .00609 9-7 .40 3 1 44 79-4 " 143-8 69.4 125.7 0199 50.4 .00606 91.2 .41 34 " 79-5 ** 144.1 44 69.6 ' 4 126.1 44 50.6 44 9 T -5 .42 38 " 79.6 ^ 144 2 .0143 69.7 .00436 126.2 .0197 5o.9 . 00600' 92.2 43 .41 44 79-7 41 144.4 69.9 126. 7 .0196 51. i .00597 92.6 44 44 .0125 %l .00381 * 4 " 70.1 " I27.O OI 9S 51-4 00595 93-i 45 47 79-8 4 * 144.5 ..0142 70.2 .00433 127.2 .0194 Si 6 .00591 93-5 .46 5 1 " 79-9 44 144-7 70.4 kt 127.4 0193 51-8 .00588 93 & 47 54 " fco.o 14 144.9 44 70.5 " I27. 7 .0192 52.0 .00585 94.2 .48 57 " 44 4 4 44 70.6 41 127.9 .0191 52-3 .00582 04 7 49 .61 " 80. i 44 145.1 .0141 70.8 .00430 128.2 52-5 44 95-i 5 .64 " 80.2 44 MS -3 44 70-9 128.4 .0190 52-7 00579 95-5 55 .80 .0124 80.4 .00378 145.6 .0140- 7'. 5 .00427 129.5 .0186 53-7 .00567 97-3 .60 97 80.7 146.2 .0139 72.1 .00424 130.6 .0183 54-6 .00558 08.9 65 '3 " 80.9 " 146.6 .0138 72.6 .00421 131.5 .0181 55-4 .00552 100.3 .70 3 .0123 81. i 00375 146.9 .0137 73.0 .00418 132.2 .0178 56.1 .00543 101.6 75 .40 81.3 '' 147-3 .0136 73-4 .00415 132.9 .0176 5-5.8 00537 102.9 .80 .62 " 81.5 44 147.6 73-8 44 133-6 .0174 57 4 .00530 104.0 .85 79 .0122 81.7 .00372 148.0 OI 35 74.1 .00411 *34-2 .0172 58.0 .00524 105.1 .90 95 81.8 148.2 .0134 74-4 .00408 134-8 .0171 58.6 .00521 106.1 3 20 GAUGING. VALUES OF b AND c, FOR THE SIX CLASSES I TO VI, P. 250, IN BAZIN'S NEW FORMULA, bit = mi, OR v = c^^mi, WHERE c(\+ -Z=} =87 Vtni, OR = 157.6 Vmi', ACCORDING AS THE \ Vml UNIT IS A METRE OR A FOOT. Value of m CLASS I, CLASS II, CLASS III, the Unit the Unit being a the Unit beinga t-he Unit being a being a Metre. Foot. Metre. Foot. Metre. Foot. Metre Foot b c b c b C b c b c b C 95 3.12 .OT22 81.9 .00372 148.4 .0134 74.7 .00408 135-' .0160 59 i .00515 107.0 i .00 3-28 44 82.0 (4 148.5 0133 75-o .00405 135-8 .0168 o -^ 59-6 .00512 107.9 1.20 3.61 3-94 .OI2I 82.4 .00369 148.9 149-3 .0132 75-4 75-9 .00402 '36.5 '37 4 .016. 60.5 61.3 .0050^ .00497 ID9.6 III .O I. 3 4.26 ' 82.6 * 149.6 .0131 76.3 .00399 138.2 .0161 62 o .00491 112.4 1.40 4-59 t< 82.8 " 150-0 44 .0160 62.6 .00488 ri 3-4 I. 5 I. 60 4.91 5-25 .OI2O 82.9 83-0 .00366 150-2 150.4 .0130 76.9 77.2 .00396 J39-3 139. .0158 .oiJ7 63.2 6?,. 8 .00482 .00479 114-5 115-6 . 1.70 5.58 " 83.1 '' 150.5 .0129 77-5 .00393 140.3 .0156 64-3 .00476 116.5 I. 80 5.91 " 83.* " 150.7 77-7 140.7 .0154 64.8 .00470 117.4 I.QO 6.24 44 83.3 44 150.9 .0128 77-9 .00390 141.1 015.^ 65.2 .00466 118.1 2.00 6.57 " 83-4 44 151-1 44 78.1 141 5 .0152 6s. 6 .00463 118.9 2.20 7.22 " 83.6 14 151-4 .0127 78.5 .00387 142.2 .0151 66.4 .00460 120.3 2.40 7.8 7 .OlTg 8 3 .7 .00^63 151-6 44 78.8 .< 142.8 .0149 67.1 .00454 121.5 2.00 2.80 8-53 9.19 t< 83.8 83-9 !< 151.8 152.0 .0126 79-i 79 4 .00384 143-3 143-9 .0148 .0147 67.7 68.2 00451 .00448 122.6 I2 3-5 3.00 9.8 4 " 84.0 " 152.2 44 79.6 ,44-2 .0146 68.7 .00445 124-4 3.20 10.50 44 84.1 u 152.4 .0125 79.8 .00381 144.6 .0145 69.2 .00442 I2 5-3 3-4 11.15 " 84.2 41 i5 2 -5 >4 80.0 144.9 .0144 ^9.6 .00439 126.0 3-6o 11.81 44 84-3 " 152-7 44 80.2 44 145-3 .0143 70.0 .00436 126.8 3-8o 12.47 0118 84.4 i5 2 -9 .0124 80.4 .00378 145.6 .0142 70.4 .00433 53:; 4 .00 4-5 13.12 14.76 84.6 T 53-3 80.9 44 146.5 .0140 7 l -5 .00430 .00427 129.5 5.00 16.4 " 84.7 44 153-4 .0123 81.2 4 375 147.0 onq 72.1 .00424 130.6 5-5 18.04 " 8 4 .8 " 153-6 .0123 8. .4 146.6 .0138 72.7 .00421 J 3i-7 6.00 19.69 " 84-9 " 53.8 4 * 81.6 41 147 8 0137 73-2 .00418 132.6 6.50 2 i-33 " 85.0 11 154.0 .0122 81.8 .00372 148.2 .0136 73-7 .00415 133-5 7.00 22.96 '* ** 44 " 82.0 148.5 0135 74.1 . OO4 I 2 T 34 .2 7-5o 23.61 " 85.1 " 1 54- 2 44 82.2 44 148.9 0134 74-5 .00408 134-9 8.00 26.25 " 85.2 " 1 54 3 44 82.3 4 149.1 44 74.8 44 I 35-5 8.50 27.89 .0117 .00356 .0121 82.4 .00369 J49-3 .0133 75-i .00405 136.0 9.00 29-53 44 85-3 u 154-5 41 82.6 M 149.6 75-4 44 *36. 5 9-5 3i.i7 " * fc " 4 * 82.7 44 149.8 .0132 75-7 .00402 i37-o o oo 32 81 44 *' 44 ** 44 82.8 44 150.0 44 7S-9 44 '37-4 I. 00 36.09 " 85-4 * 154-7 14 83-0 " 150.4 .0131 76.4 .00399 138.4 2.OO 39-37 44 85.5 4 154.9 .Ot20 83.1 .00366 150.5 .0130 76.8 .00396 139. i 3 -co 42-65 11 ** * " 83-3 44 150.9 44 77.1 T 39-7 4.00 45-93 " 85.6 1 i55-o " 83-4 44 151-1 .0129 77-4 .00393 140.2 5.00 49.21 " fc * * 4i 44 83-5 44 i5i-3 44 77-7 l * 140.7 6.00 52-49 44 B 5 .7 1 155-2 44 83.6 44 i5 T -4 .0128 78.0 .00390 141.2 7.00 55.77 " 4 .0119 83-7 .00363 151.6 44 78.3 141.8 8.00 59 -6 4i 44 * ** 4% 83.8 151.8 .0127 78.5 .0038 7 142.2 9.00 ?- 4 it 85.8 * ^55-4 44 83-9 44 152.0 44 78.7 i t 142.6 20.00 65.62 152.2 78.8 142.8 GAUGING. VALUES OF b AND v* = mi, OR v - fVmi, WHERE C\l + -~\ = S7Vmt' t OR = 157.6 \ Vm) UNIT IS A METRE OR A FOOT. ACCORDING AS THE Value of tn, CLASS IV, CLASS V, CLASS VI, the Unit being a the Unit being a the Unit being a the Unit being a Metre. Foot. Metre. Foot. Metre. Foot. Metre Fogt. 3 C b C b c b c b c b ' 05 .06 .16 0552 18.1 .01682 32-8 .0784 12.8 .02390 23.2 1015 9-9 I .03094 .02856 17.9 .07 23 .0484 20 6 01475 37-3 .0680 14.7 .02073 26.6 937 .o8 7 6 Jo. 7 11.4 .02670 19 4 20.6 .08 .26 .0461 21.7 .01405 39-3 .0644 15-5 .0196: 28.1 .0827 12. I .02521 21.9 .09 30 .0441 22.7 01345 41.1 .0613 16.3 .01868 29-5 .0786 12.7 .02396 23-0 . IO 33 .0424 23.6 .OI2Q2 42.7 .0588 17.0 .01792 30.8 075* 13-3 .02289 24.1 .11 36 .0410 24.4 .01250 44.2 .0566 17.7 .01725 32.1 .0722 1 3-9 .02201 25.2 . 12 39 0397 25-2 .OI2IO 45-6 .0547 18.3 .01667 33-2 .0696 14.4 .O2I2I 26.1 J 3 43 .0386 25.9 .01177 46.9 .0530 18.9 .01615 34-2 .0673 14.9 .02055 27.0 .14 .46 0376 26.7 01146 47-5 0515 19.4 .01570 35-i .0653 *5-3 .01990 27-7 .15 49 .0367 27.2 .01119 49-3 .0501 19.9 .01544 36.0 .0625 15-8 .01936 28.6 .16 52 359 27.8 .01094 5-3 .0489 20.4 .01490 36-9 .0618 16.2 .01884 29-3 17 56 .0352 28.4 .0107; 5i-4 .0478 20.9 oi457 37-9 .0603 16.6 .01839 30.1 .18 59 345 29.0 .01052 52.5 .0467 21.4 .0142; 38.8 .0589 17.0 -01795 30.8 .19 .62 339 29-5 .01033 53-4 .0458 21.8 .01396 39-5 577 17-3 -01759 3 T -4 .20 65 334 30.0 .oioif 54-3 .0449 22.3 .01369 40.4 0565 17.7 .01722 32.1 .21 .69 .0328 3-5 .00997 55-2 .0441 22.7 01345 41.1 0554 18.1 .01689 32.8 .22 .72 0323 3-9 .00985 56.0 0434 23-1 .01324 41.8 0544 18.4 .01658 33-7 2 3 75 .0319 3i-4 .00972 56-9 .0427 23-4 .01302 42.4 0535 18.7 .01631 33-9 .24 79 0315 3'.8 .00960 57-6 .0420 23 8 .01280 43-i .0526 19.0 .01603 34-4 25 .26 .82 Re .0310 32.2 72 6 .00945 58.3 .0414 .0408 24.2 .01262 43 .8 .0518 19.3 .01579 34-9 .27 5 .88 .0307 .0303 3.2 .0 33-o .00930 .00924 59 ** 59.8 .0403 24 5 84.8 .01229 44-9 .0502 19.9 .01528 35-5 36.0 .28 .92 .0300 33-4 .00915 60.5 .0397 25.2 .OI2IO 45-6 .0495 2O. 2 .01507 36.5 .29 95 .0297 33 7 .0090; 61 .0 0393 25-5 .01198 46.2 .0489 20.5 .01489 37-1 .30 .98 .0293 34-1 .0089; 61.8 .0388 25-8 .01183 46.8 .0482 2O-7 ; .01468 37-4 31 1.02 .0291 34-3 .00887 62.1 .0383 26.J .01167 47-4 .0476 21.0 .01450 38.0 32 05 .0288 34.7 .00878 62.9 0379 26.4 OU55 48.0 .0471 21.2 ; .01435 38.4 33 .08 .0285 35 . i j . 00869 63.6 0375 26.7 .01143 48.6 .0465 21.5 .01417 38.9 34 .12 .0283 35.41.00863 64.1 0371 26.9 .01131 48.7 .0460 21.7 \ .01402 39.3 35 15 .0280 35-7 1-00853 64-7 .0368 27.2 .O1I22 49-3 455 22. O .01387 39-9 36 .18 .0278 36.0 .00847 65.2 .0364 27.5 .OHIO 49-9 .0450 22.2 .01372 40.1 37 .21 .0276 36.3 .00841 65-7 .0361 27.7 .01100 50.2 .0446 22.4 .01360 40.6 38 2 5 .0274 36.6 .00835 66.3 0357 28.0 .01088 50.8 -0441 22.7 .01344 41.1 39 .28 .0272 36-8 00829 66.6 0354 28.2 .01079 Si-* 0437 22-9 1 .01331 41.4 .40 31 .0270 37-1 .00823 67.! 0351 28.5 .01070 S 1 -? 433 23.1 .01319 41.8 .41 34 .0268 37.4 .00817 67.7 0349 28.7 .01064 52.0 .0429 23-3 -01307 42.2 .42 38 .0266 ! 37.6 .00811 68.1 .0346 28.9 01055 52-3 .0426 23-5 . o i 298 42.5 43 .41 .0264 1 37-9 .00805 68.6 0343 29.2 .01046 52.9 .0422 23-7 .01286 42.9 44 44 0262 38-1 00799 69.1 .0340 29 4 .01036 53-2 .0418 23-9 .01274 43-3 45 47 .0261 38-4 .00796 69.6 .0338 29.6 .01030 53-6 .0415 24.1 .OI265 43-7 .46 5i .0259 38-6 .00789 69.9 0335 29.8 .01021 54-o .0412 24.3 .01256 44-o 47 54 .0258 38.8 .00786 7-5 333 30.0 .01015 54-3 .0409 24-5 .01247 44-4 .48 57 .0256 39 i .00780 70 8 0331 30.2 .01009 54-7 .0405 24.7 .01234 44-7 49 .61 .0255 39-3 .00777 71.2 .0329 3-4 .01003 55-o .0403 2 4 .8 .OI228 44 '9 50 .64 0253 39-5 .00771 7 1 5 0326 30.6 .00994 55-4 .0400 25.0 .01189 45-3 55 .80 .0247 4-5 00753 73-4 0317 31-6 .00966 57-2 .0386 25-9 .01177 46.9 .60 97 .0241 j 41.4 .00734 75-o .0308 32.5 .00939 58-9 0375 26.7 .O1143 48.3 .65 J 3 .0236] 42.3 .00719 76.6 .0300 33-3 00914 60.3 .0365 27.4 .OIII2 49 6 .70 3 .0232 43-t 00707 78.1 .0294 34-1 .00896 61.8 .0356 28.1 .01085 5-9 75 .46 .0228 43-9 .00695 79-5 .0288 34-8 .00878 63.0 .0347 28.8 .OIO58 5 2 - 1 .80 .62 .0224 44-6 .00683 80.8 .0282 .00856 64-3 .0340 29.4 .01036 53-2 85 79 .0221 45-2 .00674 81.9 .0277 36.1 00841 65.4 333 3O.O .01015 54-3 3 22 GAUGING. VALUES OF b AND c, FOR THE SIX CLASSES I TO VI, P. 250, IN BAZIN'S NEW FORMULA, bv* = mi, OR v = cVmi, WHERE 7, OR = 157.6 Vrni, ACCORDING AS THE UNIT IS A METRE OR A FOOT. ~== == &J Vml Value of /, CLASS IV, CLASS V, CLASS VI, the Unit the Unit being a the Unit being a the Unit being a being- a Mef*- Foot Foot. Metr^ Fr. Metre Foot. b e b e b c b c b c b C .90 2-95 .0218 45-9 .00663 83.1 .0273 36-7 .00829 66.5 .0327 30.6 .00997 55-4 95 3.12 .0215 46.5 .00654 84.2 .0267 37-3 00821 67.6 .0321 3 1 - 1 .00979 56.3 .00 3-28 .0213 47.0 .00649 85.1 .0265 37-8 .00795 68.5 .0316 31.6 .00963 57-2 . 10 .0208 48.0 .00634 86.9 .0258 38.8 00786 70.3 .0307 32.6 .00936 59-o .20 3-94 .0204 48.9 .00622 88.6 .0251 39-7 00765 71.9 .0299 33-5 .00911 60.7 -3 4.26 .0201 49-8 .00613 00.2 .0246 40.6 00749 73-5 .0291 34-3 .00887 62.1 .40 4-59 .0198 50.6 .00604 91.6 .0241 41.4 00734 74-9 .0285 35 l .00869 63.6 5 4.91 .0195 S I -3 .00595 92-9 .0237 42.2 .00722 76.3 .0279 35-8 .00850 64.8 .60 5-25 .0192 52.0 -00585 94.2 0233 42.9 .00710 77-7 .0274 36-5 .00835 66.2 7 5.58 .0190 52.6 .00579 95-3 .0230 43-6 .00701 79.0 .0269 37- 1 . 00820 67.2 .80 5-9 1 .0188 53-2 c 573 96.3 .0226 44-2 00689 80. i .0265 37-7 .00808 68.3 .90 6.24 .0186 53-8 .00567 97-4 .0223 44-8 00680 81.2 .0261 38.3 .00796 69.4 .00 6.57 .0184 54-3 .00561 08.4 .0221 45-3 .00674 81.7 .0257 38-9 .00784 .20 7.22 .0181 55-3 .00552 IOO.2 .O2l6 46-4 .00659 84.0 .0251 39-9 .00765 72-3 .40 7.87 .0178 56.2 .00543 101. I .0212 47-3 .00646 85.7 .0245 40.8 .00747 73-9 60 8.53 0175 .00534 103. I .0208 48.1 .00634 87.1 .0240 41.7 .00732 75- 1 .80 9.19 57-7 .00528 104.4 .0204 48-9 .00622 88.5 .0235 42-5 .00717 77 o 3.00 9.84 .0171 58.3 .00521 105.6 .O2OI 49-7 .00613 89.9 ,0231 43-3 .00705 78.4 3 .20 10.50 .0170 58-9 .00518 106.7 .0199 50.4 . 00607 91.2 .0227 44.0 .00693 79-7 3-4 11.15 .0168 59-5 .00512 107.8 .0196 51.0 .00598 92.3 .0224 44.6 .00683 80 8 3-6o 11.81 .0167 60. t .00509 108.9 .0194 .00592 93-5 .0221 45-2 .00674 81 .9 12.47 .0165 60.6 .00503 109.8 .0192 52.2 .00585 94.6 .0218 45-8 .00665 83.0 4.00 13.12 .0164 61 .0 .00500 110.5 .Oigo 52.7 -00579 95-5 .0216 46.4 .00657 84.0 4-50 14.76 .0161 62.1 .00491 112-5 .Ol86 53.9 .00567 97-6 .0210 47-6 .00639 86.2 16.40 .0159 63.0 .00485 II4.I .0182 55.0 00555 99.6 .0205 48.8 . 00624 88.3 5-50 18.04 .0157 63.8 .00479 "5-5 .0179; 56.0 .00546 101 .4 .0201 49-8 .00613 90.2 6.00 19.6^ 64.6 .00473 116.9 .0176 56.8 00536 102.9 .0197 50-7 .00600 91.8 6.50 21.33 0153 65.2 .00466 118.1 0174' 57-6 -00530 04-3 .0194 51-6 .00591 93-5 7.00 22.96 .0152 65.8 .00463 119.2 .0172: 58.3 .00524 05.6 .0191 52-3 .00582 94-7 7-50 23.61 .0151 66.4 .00460 120.3 .0170 58.9 .00518 06.7 .0189 53- .00576 96.0 8.00 26.25 .0150 66.9 .00457 121. 2 .0168! 59.5 .00512 07.8 .0186 53-7 .00567 97-3 8.50 27.89 .0149 67-4 .00454 122. 1 .o'66: 60. i .00506 08.9 .0184 54-3 .00561 98.4 9 oo 29-53 .0148 67.8 .00451 122.8 .0165 60.7 .0050- 09.9] .0182 54-9 . 00555 j 99.4 9-50 .0147 68.2 .00448 123-5 .0163: 61.2 .00497 10.8 .0180 55.6 .00549 00.7 lo.oo 32.81 .0146 68.5 .00445 124.0 .0162! 61.6 .00494 ii. 6 .oi7C 56.0 .00545 01.4 1 1 .00 36.09 .0144 69.2 .00438 125.3 .0160 62.5 .00488 13.2 .0176 57-o .00536 03.2 12. OO 39-37 0143 69.9 .00436 126.6 .0158 63.3 .00482 13.6 .0173 57-8 .00527 04.6 13.00 42.65 .0142 70.4 .00433 "7-5 .0156 63.9 .00476 13-9 .0171 58.6 .00521 05.0 14.00 45-93 .0141 70.9 .00430 128.4 .0155! 64.5 .00473 16.8 .0169 59 3 .00515 07.4 15.00 49.21 .0140 .00427 129.1 .0154! 65.1 .00470 17.5 .0167 59-9 .OC509 08.5 16.00 52-49 .0139 71.7 .00424 129.8 .0152 65.6 .00463 18.8 .0165 60.5 .00503 09.6 17.00 55-77 72.1 " 130.5 .0151 66.1 .00460 19.7 .0164 61.1 .00500 10.7 18.00 59.06 .0138 72.5 .00421 131." .0150; 66.6 .00457 20. t .0162 61.6 .00494 ii. 6 19.00 62.34 .0137 72 8 .00418 131.1 .0149 67.0 -00454 21.4 .0161 62.1 .00491 12.5 20.00 65.62 132.2 .0148 67.3 .00451 21.5 62.5 13-2 GAUGING. 323 << O" J ^S^'S.^S&^SS f.oS'^^^^^SS- 55 O fc W 1 5J 'S.^^^oSS 1 ^"^ ^^S^R^^^So H S S 3 O ^sr^^^cSo^?^ ^^s-^e^^^^^R W o 2 ; H U ^RRo8oo'^ 5- So'RJ^cSSc^O 00* s trl s s^^-^o"^ o^o 2" ^^^^^^5,^0^?: Pi < VO fc %> V Ococo w>i-i OO O coco O Oco -i- o r^ ro t^ O O c s 11 8 ^o^^^^cS^s^^ ^^s^^gjas^i: ^ H u fc < a 4J So^i fe 3 "S 'c tD g mOOOOOOOOO r>OOOOOOOOO OiHCMcomOOOOO Oi-iCMcotoOOOOO u s M ) | V J >. J D J = S & io- = u 10- u 3 2 4 GAUGING. "> "*s COUIOOr^rtioOCOl^O COTfT^-MOl-iOioOO* (Li MI-IMI-IM M M M M OO cooo N oo ^J-oo O ""> -f-OOO^O uioo M o C^*}T}-Tf-"">u-)OOt^r->. NcocO'1-'^-ir>ir>\r>OO O o tu V ^ o u oS s I s s o o s o .0 _ . _ o u u M en vo Q^O C4 10 co ^t~ w * co vo O 1 * o c^ in oo r c< t t COvO O^O O^ M COO O^ O O^ dJ mOOOOOOOCO u->OOOOOOOOO u O w CM co u~> OOOOO 0*^04 co to O O O O O rt xi >0 GAUGING. 325 < 8 I eooooONNjOMO r>ooui^cc>roor^ S pq 3- j coNr^M\r>O>^>r-^Ou^ ^coc^ioococoMtoco H s S D I coOOOcOMMr^CJHi ir>cOO"^cOOO'^cOO co^^u^or^ocooo C4coTt-rj-mvoor^r^co E H Metre. co M r^ M \r> o uico M \o ^co w w> o coco M co o MC'4NCOCO'<3 - '^ r 1 - "">'> M IH W W N CO' W 1" ^ <5T c* I MCOr^^-WMC^r^.TtcO ^HioocOHiOHivOON coco^u^oi^OcooO Mcoco^t'inOr^r^.coo M r be c M co M r^ co r^ HI rf co co o M TJ- M ^ 5 %j II ?j r 40,000 o Sfa3S5R**S& ? ^5 5SR-R53: H & w S OOxnO^OOOcoO coo O T t- co O d ^N D _0 c/5 S < V J3 o r-^cocoo N U^MCO -j- w r-^w OO w co u-, o fa o fc hH M CO S ^O-fO-^O r-~ ^- m e< ^o^^r^^o ^>^u> W s S 5 ^ ?S*RSRffS& 8S*WSsaS Q c*"i c^ co o c^ rt ho vO co O co ^ co O T O O O cn\o co T** oo o ^ *-H CO vO O^O C^ u^ CO ^t" W M COO Q^^O C* ID CO ^" C^ ^ 5 "o fc. M coo oo o M cno o^o o^ fa V ^3 '5 coco cocoO"^c^O OMO coco in coio in '$ w EC H 1 M ^-co M -f O coo O ** OWOcOMinOMOM tf O (b vttttzssszs ffa&s%!?s5^. . r*" Is be c 2 1 s^^g^ss^^;? -s^ss&RRg. Vj II O co ^t" HH O CJ c^ w t^ O u^ T}- O *^* coo *o Tj* O ^ co < u cor- coo-tr-N r- oo M mco O m o coco co D ft sS 2g u H I O M O rf N co N Oco M co O m M o mo M M co D f w Metre. -s^sfr;^^ -cstasRSg* is i *O W O^ ^ W CO CO C^ H4 M CO CO U~> Q\ *f U") O CO "1" Tf" Q u 0^0 Ocoa^O mo 1-0 -^0 OinM in M < s W 1 rf- o 1"^ co O co xr> \o O^ co M O *^" r^ r}~ i^ oo r^ O^ t*** M hH D O Metre. -sssasrassj? --j?^?^as?s O rt O COO CO rj-co O Tj- O O O COO co -3-OO O "* O O M coo OO M moo rt M M coo OO N mco -^- M & 5 "o 1 to M too OO O M coo OO O v> "rt "c ED K mOOOOOCOOO mOOOOOOOOO QMC^comOOOOO QMNcomOOOOO O 4) u MMcomin Mwcomin M ^ . ^ j 1 1^ So* ^= tt ot*o' ^^ M GAUGING. MANNING'S VALUES OF c IN THE FORMULA v UNIT BEING A METRE OR A FOOT. Value of m. Very Smooth Surface. Smooth Surface. Surface not very Smooth. Rough Surface. Metre. Foot. Metre. Foot. Metre. ' Foot. Metre. Foot. Metre. Foot. 05 .16 61 no 47 85 36 65 30 54 . IO 33 68 123 52 94 40 72 34 62 .20 .66 76 137 59 107 45 81 3S 69 30 .98 82 I 4 8 63 114 48 87 41 74 50 1.64 89 161 69. 125 52 94 45 81 I .OO 3-28 TOO 181 77 "39 59 107 50 9i 2.OO 6.56 112 203 86 156 66 119 56 IOI 3.00 9.84 120 217 92 167 71 128 60 109 5-oo 16.40 131 237 IOI 183 77 139 65 118 15.00 49.20 157 284 121 219 92 167 79 143 Value of m. Surface in Earth. Surface in Gravelly Soil. Irregular Surface. Very Irregular Surface. Metre. Foot. Metre. Foot. Metre. Foot. Metre. Foot. Metre. Foot. 05 .16 24 41 20 36 17 31 15 27 .IO -33 27 49 23 42 19 34 17 31 .20 .66 31 56 25 45 22 40 19 34 30 .98 33 60 27 49 23 42 2O 36 50 1.64 36 65 30 54 25 45 22 40 I.OO 3-28 40 72 33 60 29 53 25 45 2.OO 6.56 45 81 37 67 32 58 28 5i 3-oo 9.84 48 87 40 72 34 62 30 54 5.00 16.40 52 94 44 79 37 67 33 60 15.00 49.20 63 114 52 94 45 81 63 114 328 EXAMPLES. EXAMPLES. 1. What fall must be given to a canal 2600 ft. fong, 7 ft. wide at the top, 3 ft. wide at the bottom, i| ft. deep, and conveying 40 cu. ft. of water per second ? (/ = *V) Ans. i in 135. 2. Determine the fall of a canal 1500 ft. long, of 2 ft. lower, 8 ft. upper breadth, and 4 ft. deep, which is to convey 70 cu. ft. of water per second? (/= .008.) Arts, i in 1088.4. 3. For a distance of 300 ft. a brook' with a mean water perimeter of 40 ft. has a fall of 9.6 ins. ; the area of the upper transverse profile is 70 sq. ft., that of the lower 60 sq. ft. Find the discharge. (/= .008.) Ans. 352.12 cu. ft. per sec. 4. In a horizontal trench 5 ft. broad and 800 ft. long it is desired to carry off 20 cu. ft. discharge and to let it flow in at a depth of 2 ft. ; what must be the depth at the end of the canal ? (/ .008.) Ans. 1.36 ft. 5. Water flows along an open channel 12 ft. wide and 4 ft. deep, at the rate of 2 ft. per second. What is the fall? A dam 12 ft. by 3 ft. high is formed across the channel ; how high will the water rise over the crest of the dam ? Ans. i in 480, / being .08 ; 1.899 ft- 6. A stream is rectangular in section, 12 ft. wide, 4 ft. deep, and falls i in loo. Determine the discharge (i) with an air-perimeter; (2) without air-perimeter. (/ = .008.) Ans. (i) 646 cu. ft. per sec. (2) 665.088 cu. ft. per sec. 7. A canal 20 ft. wide at the bottom and having side slopes of i| to i has 8 ft. of water in it ; find the hydraulic mean depth. Ans. 5.163 ft. 8. The water in a semicircular channel of 10 ft. radius when full flows with a velocity of 2 ft. per second; the fall is i in 400. Find the coefficient of friction. Ans. .2. 9. Calculate the flow per minute across a given section of a rectan- gular canal 20 ft. deep, 45 ft. wide, the slope of the bed being 22 ins. per mile and the coefficient of friction per square foot = .008. Ans. 292.856 cu. ft. 10. Why does the water of a river rise on the formation of the ice ? 11. Find the depth and width of a rectangular stream of 900 sq. ft. sectional area, so that the flow might be a maximum ; also find the flow, f being .008 and the slope 22 ins. per mile. Ans. 21.21 ft. ; 42.42 ft. ; 4885 cu. ft. per sec. 12. The section of an aqueduct is a trapezium with a bottom width of 6.56 ft., a top width of 7. 546 .ft., and a depth of 7.874 ft., the slope is 6 per 1000, and the faces of the aqueduct are of brickwork. Determine EXAMPLES. 329 the discharge in cubic feet per second when the depth of the water is 4.92 ft., using the coefficient given by (a) Bazin ; (b) Kutter ; (c) Man- ntng. Ans. (a) 471.276 ; (b) 494.5484 ; (c) 487.6973. 13. An aqueduct of rectangular section is to convey 9504 (Imp.) gal- lons of water per hour at the maximum velocity of flow. Assuming as a first approximation that b = .000114, and tnat the slope is .33 per 1000, find the proper width and slope. Also find the corresponding velocity of flow. Ans. i. 01 ft. and 3 in 10,000; .828 ft. per sec. 14. What head is required to give a velocity 4 ft. per second in a semicircular channel of 3 ft. diameter and 5000 ft. long,/ being .0064? Ans. 10^ ft. i 5. The section of a length' of La Roche Canal in rock has a bottom width of .7 m., one vertical face and the other face inclined to the hori- zen at an angle tan" 1 2. The mean velocity of flow, when the water runs .5 m. deep, is .514 m. per second. Find the slope, a suitable value for the coefficient b or c being selected from the Tables. Ans. .002. 16. A section of the La Roche Canal in earthwork has its sides sloped at 45 and has a bottom width of .3 m. When the depth of the water is 0.5 m. the discharge is at the rate of 205 litres per second. Determine the slope, a suitable value for the coefficient b being selected from the Tables. Also show that, according to Bazin's formula, the maximum surface and the bottom filament velocities are .816 m. and ,49 m., respectively. Ans. Slope = .002. 17. Water flows along a symmetrical channel, 20 ft. wide at top and & ft. wide at bottom ; the friction at the sides varies as the square of the velocity, and is i Ib. per square foot for a velocity of 16 ft. per second. Find the proper slope so that the water may flow at the rate of 2 ft. per second when its depth is 6 ft. Ans. i in 3445. 1 8. Calculate the flow across the vertical section of a stream 4 ft. deep, 18 ft. wide at top, 6 ft. wide at bottom, the slope of the surface being 18 in. per mile, (f = .008.) Ans. 110.9376 cu. ft. per sec. 19. The waterway in a channel of a regular trapezoidal section, has a sectional area of TOO sq. ft. If the banks slope at 40 to the horizontal, what will be the best dimensions for the section ? Ans. Bottom width = 5.25 ft. ; depth of water =' 7.22 ft. 20. The sides of an open channel of given inclination slope at 45 and the bottom width is 20 ft. Find the depth of water which will make the velocity of flow across a vertical section a maximum. Ans. 6.73 ft. 21. The banks of a channel slope at 45; the flow across a transverse section is to be at the rate of loo'cubic feet at a maximum velocity of 5 ft. per second. Determine the dimensions of the transverse profile. Ans. 11.05 ft. wide at bottom ; 2.28 ft. deep. 22. What dimensions must be given to the transverse profile of a 33 EXAMPLES. canal whose banks slope at 40, and which has to conduct away 75 cubic feet with a mean velocity of 3 ft. per second? Ans. Depth 3.6 ft.; width at bottom = 2.62 ft. 23. The section of a canal is a regular trapezoid ; its slope is i in 500; its width at the bottom is 8 ft.; the sides are inclined at 30 to the vertical. On one occasion when the water was 4 ft. deep a wind was blowing up the canal, causing an air-resistance for each unit of free sur- face equal to one fifth of that for like units at the bottom and sides, where the coefficient of friction may be taken to be .08. Determine the discharge. Ans. 75.34 cu. ft. per sec. 24. A canal is 20 ft. wide at the bottom, its side slopes are i| to i, its longitudinal slope is i in 360; calculate H.M.D. and the flow per minute across any given vertical section when there is a depth of 8 ft. of water in the canal. (Coeff. of friction = .008.) If a weir 2 ft. high were built across the canal, what would be the increase in the depth of the water ? Ans. 5.24 ft.; 2762.7776 cu. ft. per sec.; 2.79 ft. 25. In the Ourcq canal the earthen banks slope at cot~ J i|, and the bottom width is 3.5. Find the depth of the water when the discharge is 3000 litres per second, the slope of the canal being .1236 per 1000. Also find the mean velocity. Ans. 1.5 m. to 1.4 m.; .4 m. per sec. 26. The banks of a canal slope at 45, the section being a trapezium. The discharge is to be 1200 litres per second at the rate of .5 m. per second. Find the best bottom width and depth and also the slope. Ans. .94 m.; 1.14 m.; .0004 according to Bazin and .0003 accord- ing to Manning, the mean being .00035. 27. In the transverse section ABCD of an open channel with a ver- tical slope of i in 300, the bottom width is 20 ft., the angle ABC = .90 and the angle BCD = 45. Find the height to which the water will rise so that the velocity of flow may be a maximum ; also find the dis- charge across the section, f being .008. Ans. 11.715 ft.; 1584 cu. ft. per second. 28. The sewers in Vancouver are square in section and are laid with one diagonal vertical. To what height should the water rise so that (a) the velocity of flow may be a maximum; (b~) the discharge may be a maximum? (A side of the square 12 in.) Ans. (a) .292 ft. above horizontal diameter. (<*) 5797 ft. " 29. The section of a channel is a rhombus with a diagonal vertical. How high must the water rise in the channel (a) to give a maximum of flow, and (ff) to give a maximum discharge? Ans. .If D is the length of the horizontal diameter, and if is the inclination of a side to the vertical, the water must rise above the horizontal diameter to the height D cot x .207 in (a} and to the height D cot 6 x .4099 in (ft). 30. An aqueduct has a given slope and a square section with a diag- EXAMPLES. 33 r onal vertical. Show that the discharge at maximum velocity, the dis- charge when running full and the maximum discharge are in the ratios of i to 1. 115 to 1.140, and that the corresponding mean depths are .293^, .25^, and .27^, a being a side of the square. 31. An aqueduct, with a section in the form of an isosceles right- angled triangle of height h, is laid with its base horizontal. Compare the quantities of water conveyed (a) when running full ; (&) when the velocity is a maximum; (Q*. 37. A circular aqueduct of 6.56 ft. diam. conveys 49.44 cu. ft, of water per sec. The slope is i in 10,000. Find (a) the angle subtended at the centre by the water-line ; (b) the clear head above the water sur- face ; (c) the velocity of flow. Ans. (a) 240 30' ; ( b] 1.63 ft.; (c) 1.815 ft- P er sec - 38. The Avre circular aqueduct conveys 2.05 cm. per second, and in one length the slope is 4 in 10,000. Its water-line subtends 120 at the centre. Find the radius, taking b .0002 as a first approximation. The surface has a very smooth coat of cement .02 in. thick; de- termine the actual waterway, the wetted perimeter, the mean depth, the velocity of flow, and the clear height above the water-line. Ans. Radius = .88 m. ; 1813 sq. m. ; 3.549 m. , .51 m. ; 1.13 m. per sec. ; .445 m. 39. The Potomac aqueduct, which is faced with brick, has a diameter of 9.0225 ft. and a slope, of .143 in 10,000. The water-line subtends an angle of 240 at the centre. Taking b = .0000609, determine quantity of water conveyed in gallons per day. Ans. 69,997,071 Imp. gallons. 84,019,066 U. S. gallons. 40. Taking b = .0000609, find the angle subtended at the centre by the water-line and also find the free height above the water-surface in the Vanne aqueduct when conveying 49.442 cu. ft. per second, the diameter of the aqueduct being 6.562 ft., and the slope i in 10,000. Ans. 240 30'. 41. Show that the quantities of water conveyed by a circular aque- duct of radius r, when the water-line subtends an angle of 240 at the centre, when the velocity of flow is greatest, when running full, and when the quantity conveyed is a maximum, are in the ratios of i to 1.086 to 1.131 to 1.188, and find the angles subtended at the centre by the water-lines in the three last cases. Also determine the mean hydraulic depths. Ans. Angles, 257 27' ; 360 ; 308. Mean depths, .603^ ; .6o8r; .5^; .573^. 42. For a small tachometer the velocities are .163, .205, .298, .366, .61 metre ; the number of revolutions per second are .6, .835, 1.467, 1.805, 3- l 4 2 - Find the constants corresponding to the wheel. Ans. .169; .061. 43. Assuming (i) that a river flows over a bed of uniform resistance to source; (2) that to maintain stability the Velocity is constant from source to mouth; (3) that the river sections at all points are similar ; (4) that the discharge increases uniformly in consequence of the supply from affluents determine the longitudinal section of such a river. Ans. A parabola. EXAMPLES. 333 44. In an aqueduct with a slope of i in 10,000, the depth of water corresponding to a condition of uniform steady motion is 1.77 ft. At a certain point the depth is increased to 4.43 ft. by a weir 3.77 ft. in height. Find the distance to which the "rise" extends along the aqueduct. Ans. 50,038 ft. 45. The channel of a river 328 ft. wide is narrowed by the abutments of a bridge to a width of 42.65 ft. The depth of the water under the bridge is 12.63 &> an< ^ tne quantity of flow per hour is 2,406.250 gallons. Find the height of swell. Ans. .104 ft. 46. In a broad channel of approximately rectangular section there is a small change of n% in the depth. Show that -the corresponding changes in the velocity of flow and in the discharge are \n% and \\n% respectively. Also, if the banks slope at. an angle 6, show that the nhvl b i \ , nhQl ^b i \ changes become - -- - T - and - : - respec- 100 \zA P sin 6/ 100 \2A P sin 6/ tively, /i, b, A, P, v, and Q being the initial depth, breadth, area of water- way, wetted perimeter, velocity of flow, and discharge, respectively. CHAPTER IV. RAMS, PRESSES, ACCUMULATORS, WATER-PRESSURE ENGINES. I. Hydraulic Rams. By means of the hydraulic ram a quantity of water falling through a vertical distance h v is made to force a smaller weight of water to a higher level. The water is brought from a reservoir through a supply- pipe 5. At the end of this pipe there is a valve opening into an air-chamber C , which is connected with a discharge-pipe D. At E there is a weighted check- or clack-valve opening inwards, and the length of its stem (or the stroke) is regulated by means of a nut or cottar. When the waste-valve at E is open the water begins to escape with a velocity due to the head // L and suddenly closes the valve. The momentum of the water in the pipe opens the valve at B, and a por- tion of the water is dis- charged into the air- vessel. From this vessel it passes into the discharge-pipe in consequence of the reac- tion of the compressed air. At the end of a very short interval of time the mo- FIG. 179. mentum of the water has been destroyed, the valve opening into the chamber C closes, the waste-valve again opens, and the action commences as before. It is found that 334 HYDRAULIC PRESS. 335 the efficiency of the ram is increased by introducing the small air-vessel F. The wave-motion started up in the supply-pipe by the opening and closing of the valve opening into the chamber C, has been utilized in driving a piston so as to pump up water from some independent source. Let v be the velocity of flow in the supply-pipe at the moment when the valve at E is closed. Let W l be the weight of the mass of water in motion. W v 3 Then - L is the energy of the mass, and this energy is expended in opening the valve at B, forcing the water into the air-chamber, compressing the air, and finally causing the elevation of a weight W 2 of the water through a vertical dis- tance h' . Let hj be the head consumed in frictional and other hydraulic resistances. Then W v z W 2 (h f + /y) = the actual work done = - . This equation shows that, however great // may be, W a has a definite and positive value, and therefore water may be raised to any required height by the hydraulic ram. WJt' The efficiency of the machine = * . , and may be as much w \ ll \ as 66 per cent if the machine is well made. According to d'Aubuisson, 2. Hydraulic Press. The hydraulic press is a machine by means of which great pressures can be exerted and heavy weights lifted, the energy being transmitted through water. It consists essentially of a strong cast-iron or cast-steel cham- ber or cylinder containing a plunger or ram which is acted 336 HYDRAULIC PRESS. upon by water pumped through piping into the chamber by a single-acting force-pump, which may be either worked by hand or by power. The action of the press depends on the principle that fluids press equally in all directions and thus the pressure per square inch on the ram is equal to the pressure per square inch on the pump-plunger. Origi- nally discovered by Pascal, FIG. 180. the press was first made of practical utility by Bramah, who made the moving parts water- tight by the introduction of cup-leather packing. The ram is packed with a leather collar of f| form which is fitted into a recess turned out in the neck of the cylinder and is kept in place by the cylinder-cover gland. According to experiments made by Hick, the friction at the collar increases directly with the diameter of the ram and with the FlG - pressure, but is independent of the depth of the collar, law of friction is expressed by the following formula: Hick's the total frictional resistance = .0314^ or .0471^, according as the leather is in good condition and well lubri- cated or is new and badly lubricated. The friction is about I per cent of the pressure for a 4-in. ram. At low pressures hemp packing is invariably used, and sometimes also for pressures as great as 2000 Ibs. per sq. in., but, generally speaking, it is rarely used for pressures exceed- ing about 700 Ibs. per sq. in. The ram is driven forwards by the pressure of the water through the tight collar, and is capable of lifting a weight or exerting a pressure which is limited in HYDRAULIC PRESS. 337 magnitude only by the strength of the chamber and connec- tiogs and by the capacity of the pump. Let L be the stroke of the ram. Let W be the weight on the ram, including the weight of the ram. Then the work done = Let Q be .the axial force on the plunger produced by a force P on the pump-lever at a distance p from the fulcrum. Let q be the distance between the fulcrum and the axis of the plunger. Then, disregarding fluid friction, the friction at the fulcrum, and the leather or ' ' packing ' ' friction, Pp = 07. irD* *D> W D' Pp But ^^-W- If r l , r are the internal and external radii of a press, and if p^ , p QJ arid ,/are the intensities of pressure at the internal and external surfaces and the intensity of stress at the radius r, then f_Pf* *"* so' o jf\' i , SQ r\ ' o' i r o r i r ~ ; V r \ (See Appendix, "Th. of Structures," Bovey.) Hydraulic presses of different designs, but which are all more or less modifications of the Bramah, are employed for a variety of pressing and lifting operations. For example, they are used in making lead pipes, in expressing oil from seeds, in baling cotton, in pressing yarn, in packing hay, etc., while the modern systems of punching, riveting, stamping, forging. 338 HYDRAULIC JACK. shearing, welding, and bending depend upon the peculiar advantages of hydraulic power for such purposes,* Hydraulic presses for forging have largely superseded the steam-hammer. FIG. 182. Hydraulic Press. FIG. 183. Portable Riveter. FIG. 184. Baling-press. and it is now common to find presses with capacities ranging from 4000 to 10,000 tons, the working intensity of pressure being as great as 3 tons per sq. in. The hydraulic jack, Fig. 185, is a portable machine for raising heavy weights through short distances. It is a com- pact combination of a force-pump and a press. The ram Q fits the press 5" and is made water-tight by the cup-leather D. The pump is worked by the up-and-down movement of a lever which presses upon a cam or is connected with other suitable gearing and communicates motion to the pump-plunger .R. The water in the chamber is thus forced through a valve into the hydraulic cylinder, developing a pressure which causes the ram to rise and to lift the load resting on the head H. * In America compressed air is largely used for punching, riveting, etc. ACCUMULATOR. 339 As the pump-plunger rises a partial vacuum is produced in the pump-chamber, and the pressure in the reservoir B overcomes the resistance of the spring on the inlet-valve and opens a passage for the water into the pump-cham- ber. To lower the jack, a relief- valve is unscrewed, and the water returns to the reservoir B while the ram falls. The ram FIG. 185. FIG. 186. may be prevented from turning round by .means of a steel set- pin screwed on the side of the press and fitting a vertical slot in the ram. The construction and action of the punching-bear, Fig. 1 86, are essentially the same as in the hydraulic jack. By actuating the lever Z, the water passes into the hydraulic cylinder C and by its action forces the punch P down. The punch is raised by first opening a relief-valve and then lower- ing the leVer M, which causes the cam to raise the hydraulic ram, and the water from the hydraulic cylinder flows back into the reservoir. The relief-valve is now closed and the punching operation may be again repeated. 3. Accumulator. Low pressures of 170 Ibs. (= 392 ft.) to 250 Ibs. (= 576 ft.) per sq. in. can sometimes be obtained from a natural supply or from a reservoir, but the higher 34 ACCUMULATOR. pressures of 700 Ibs. (= 1612 ft.) to 1000 Ibs. (= 2304 ft.) per sq. in. and upwards, which are almost exclusively adapted to the working of intermittent machines, must be artificially produced by means of pumping-engines. In a direct supply the capacity of these engines must be sufficient to meet the maximum demand at any moment, but the fluctuation in the demand upon the mains for cranes, capstans, elevators, etc., was soon found to be so great as to render imperative some method of storing energy. This has been effected by the introduction of the accumulator, which, in its simplest form, consists of an annular cylinder (Fig. 187) partially or wholly filled with scrap, slag, or other heavy material, or of a series of trays (Fig. 188) loaded with pig iron or lead, supported by a cross-head on the top of a ram working in a cylinder with a FIG. 187. FIG. 188. stuffing-box and gland at the upper end. The pressure -water is admitted by a branch pipe at the lower end and raises the ram together with the weight it carries. Thus, if JKtons are .lifted through a vertical distance s and if the water -pressure on INTENSIFIED 34 t the ram of d in. diameter is p Ibs. per sq. in., the total store of energy in foot-pounds = 224011 s = --- sp. 4 When the accumulator has reached the highest point it actuates a lever which shuts off the steam so that the engines, cease to work and the accumulator falls. When it has reached the lowest point it again actuates a lever which opens a valve and admits steam. The engines again commence to work and the accumulator rises. In small plants the accumulator fully provides for the storage of sufficient energy to meet the momentary fluctuations of demand for the power necessary to work machines which are intermittent in action, and without the accumulator pump- ing-engines of greater capacity would be required. In large plants, as in the cities of London, Manchester, and Glasgow, the total accumulator storage capacity is a very small fraction of the total supply, and at the times when the demand is heavy the accumulators are usually almost stationary. In such cases they may be considered rather as regulators of pressure. They are also of great importance in automatically facilitating the control of the plant, and act as buffers in preventing break- age and shocks. If lack of space prevents the use of an accu- mulator of the type just described, an intensifier, Fig. 189, may be employed. Water at a pressure of p Ibs. per sq. in. is admitted from the water-mains or from a tank at a suitable elevation to the lower side of a piston of diameter D ins., work- ing in an hydraulic cylinder. The piston-rod of diameter d ins. forms the ram of the accumulator B, and works through a water-tight neck. Thus the pressure in the accumulator in Ibs. per sq. in. TlD* 342 DIFFERENTIAL ACCUMULATOR. and this is also the intensity of the pressure in the hydraulic mains C. Tweddell's differential accumulator. Fig. 190, is also designed for cases in which space is of importance. A heavy cylinder A, with the usual glands and cup-leathers at the top FIG. 189. FIG. 190. and bottom, is loaded with a number of lead or cast-iron weights W, fitted into each other, and slides upon a ram />, fixed at the upper end by a bracket and at the lower by a step. A brass liner is shrunk upon the lower portion of the ram* so that its diameter is slightly greater than that of the upper portion. A hollow passage C is drilled axially along the ram and connects with a cross-passage just above the brass liner. The water is pumped through the inlet-pipe /, fills these passages and exerts an upward pressure over an effective area equal to the difference between the areas of the lower and upper portions of the ram. Thus very heavy pressures, up to 2OOO Ibs. per sq. in., or more, can be readily obtained with a comparatively small weight. But the volume of water is * The ram, however, is usually solid steel. DIFFERENTIAL ACCUMULATOR. 343 small, and any large demand for power will cause the loaded cylinder to fall rapidly, so that when it is brought to rest a considerable increase of pressure ;is developed which is of advantage in punching, riveting, etc. The uppermost weight is connected by means of a chain with a relief-valve which enables the limiting positions of the cylinder to be automati- cally regulated. Let J^be the total dead weight lifted. Let F be the friction of each of the cup-leathers. Let d v , d 2 be the diameters of the lower and upper portions of the ram. With the cylinder at the height x above its lowest position, let p v be the intensity of pressure in the inlet-pipe / when the cylinder is rising, and p. 2 the intensity when it is falling. Then W+ 2F A = w *+ ~ - W- 2F A = '*+ ~W - O Hence an approximate measure of the variation of the intensity of pressure is i6F Pi " p ' = ^d I -d,y and the value of this variation is ordinarily from about I per cent of the pressure for a i6-in. ram to about 4 per cent for a 4-in. ram. Experiment has shown the efficiency of an accumulator to be as high as 98 per cent, I per cent being lost in charging and i per cent in discharging. Its total store of energy is comparatively small and it cannot maintain a supply for any length of time, but it possesses the great advantage of being- able to use its energy at a high rate for a short period. 344 HYDRAULIC ENGINES. Fig. 191 represents a convenient form of accumulator known as Brown's Steam Accumulator. A ram R works in the hydraulic chamber //, into which water is forced by a pair of engines. A piston P is attached to the upper end of the ram and works in, a cylinder supplied with steam direct from the boilers. As soon as pressure- water is supplied to hydraulic ma- chinery the ram and piston fall, opening the steam-port, so that steam passes into the engine-cylinders. The pumps then commence to work and force in FIG. 191. more water to replace that which is being drawn off. This accumulator is specially for use on ships. 4. Water-pressure Engines. In these engines water under pressure is admitted into a strong chamber or cylinder, and acts upon a piston or plunger in precisely the same manner as in the case of the steam-engine. The cylinder is made of gun -metal or of cast iron, and its thickness /, which is relatively large on account of the wear, may be calculated from the formula /ins. = .0024/V/+ 1.25 ins., f a being the pressure in atmospheres, and d the diameter in inches. The frictional resistances and the possibility of severe shocks are increased by rapid motion and reversals of motion. Hence the velocity of flow in the supply-pipe should not exceed 10 ft. FIG. 192. HYDRAULIC LIFTS. 34S per second, and preferably should be limited to 6 ft. per second (Art. n, p. 156), while the plunger should have a long stroke. In practice the stroke is usually from 2\ to 6 times FIG. 193. Sectional Elevation. FIG. 194. Cross-section. FIG. 195. Freight-hoist. FIG. 196. Balanced-ram Lift. the diameter of the cylinder, and the mean velocity of the plunger is about I ft. per second, rarely exceeding 80 ft. per minute. As the water is practically incompressible, its free and immediate passage should be insured by means of large 346 HYDRAULIC LIFTS. and wide-open ports. An important advantage connected with this property of incompressibility is that the hydraulic resistances may be indefinitely increased by simply closing a valve. Thus no brakes are required, but the water contains within itself its own brake, and an absolute control is provided which secures the highest degree of safety. The water-pressure engine is necessarily a slow-moving machine, and is both cumbrous and costly unless actuated by pressures of great intensity. These engines are advantageously employed in working cranes, hoists, elevators, capstans, dock- gates, presses, and other machinery in which the action is of an intermittent character. The hydraulic-ram lift, Fig. 197, more completely utilizes than any other the properties of incompressibility and direct pres- sure, and, owing to its greater safety, its adoption is sometimes recommended for elevators of con- siderable height. Under a full load its efficiency may be as great as 95 per cent. The speed of a suspended lift is rarely less than 100 ft. per minute and often ex- ceeds 500 or 600 ft. per minute. Between such limits a large varia- tion in the efficiency might be expected, and although the effi- ciency under a full load, even when the ram-stroke is multiplied 8 or 10 times, may be 75 or 8 per cent, it may also fall below 40 per cent when the load is light. The chief loss of efficiency is FlG< IQ7< due to the fact that the same quantity of pressure-water, and therefore of energy, is .used HYDRAULIC ENGINE. 347 \vhether the load is heavy or. light. Various, devices have been .adopted to remedy this evil : the length of stroke may be automatically proportioned, as in the Hastie engine, to the work to be done; the pressure-water may be admitted for a part of the stroke only, the remainder being provided by the discharge-water; cranes and elevators are often provided with a large cylinder for heavy loads and a small cylinder for light loads, and for the same purpose a single cylinder with a differ- ential piston is sometimes used. Other important losses of efficiency are due to (a) pipe friction; (b) elbows, curves, etc., and abrupt changes of sec- tion ; (c) the friction of mechanism. Let p m be the mean intensity of the pressure in the cylinder. Let j- be the stroke. Let i' m be the mean velocity of the plunger. Then the work done per stroke = p m s \ 4 the quantity of motive water used per stroke nd~ I nd* according as the engine is of the double- or single-acting type. Analysis. In a direct-acting pressure-engine let A be the sectional area of the working cylinder (Fig. Let a be the sectional area of the supply- pipe. Let A = na. FlG ' I98 ' Let Wbe the weight of the water, piston, and other recip- rocating parts in the working cylinder. Let / be the length of the supply-pipe. Let f be the acceleration of the piston. Then nf is the acceleration of the water in the supply-pipe. 348 HYDRAULIC ENGINE. The loree required to accelerate the piston - & and the corresponding pressure in feet of water _^/ ~ w A g ' The force required to accelerate the water in the supply pipe wal and the corresponding pressure in feet of water A Similarly, if I' is the length of the discharge-pipe and . its sectional area, the pressure-head due to the inertia of the discharge-water Hence the total pressure in feet of water required to over- come inertia in the supply-pipe and cylinder W The quantity A -\- nl has been designated the length of iv A working cylinder equivalent to the inertia of the moving parts. Let the engine drive a crank of radius r, and assume that the velocity V of the crank-pin is approximately constant. Then the acceleration of the plunger when it is at a distance x from its central position HYDRAULIC ENGINE. 349 and the pressure due to inertia Let i' be the velocity of the plunger in the working cylinder. Let u be the velocity of the water in the supply-pipe. Let // be the vertical distance between the accumulator-ram and the motor. Let / be the unit pressure at the accumulator-ram. Let / be the unit pressure in the working cylinder. Then PQ u 2 p i' 2 ( losses due to friction, sudden w~^ 2g ~ w~^~ 2g -*~\ changes of section, etc. Thus / 2 " W 2g - // + losses. 7/ 2 U 2 The term ---- + losses may be approximately expressed 7/2 in the form K , K being the coefficient of hydraulic resist- ance. Hence the term h being disregarded, as it. is usually very small as compared with . Thus the total pressure-head in feet required to overcome inertia and the hydraulic resistances HYDRAULIC ENGINE. and is represented by the ordinate between the parabola ced and the line ab in Fig. 199, in which afgb is a rectangle, ab representing the stroke 2r, the pressure due to inertia at the end of the stroke, and oe = K the pressure required to overcome the hydraulic resistances at the centre of the stroke. FIG. 199. The ordinate between the parabola fmg and the line fg represents the back pressure, which is necessarily proportional V' 2 to the square of the piston-velocity, i.e., to -~^(r 2 x*}. Hence the effective pressure-head on the piston, transmitted to the crank-pin, is represented by the ordinate between the curves fmg and ced. The diagram shows that the pressure at the end of the stroke is very large and may become excessive. It is therefore usual to introduce relief-valves or air-vessels ta prevent violent shocks. In certain cases, however, as, e.g., in a riveting-machine, a heavy pressure at the end of the LOSSES OF ENtKGY 4N HYDRAULIC ENGINES, ETC. 35 r stroke, just where it is most needed to close the rivet, is of great advantage, and therefore the inertia effect is increased by the use of a supply-pipe of small diameter and an accumu- lator with a small water section (Fig. 197). By equation (i), This speed v can be regulated at will by the turning of a cock, as in this manner the hydraulic resistances may be indefinitely increased. Let the engine be working steadily under a pressure P, and let V Q be the speed of steady motion. Then and c useful resistance overcome by the piston ( -J- friction between piston and accumulator-cylinder. If P is diminished, the speed V will be slightly increased, but in no case can it exceed \ / --~~. V wK 5. Losses of Energy. The losses may be enumerated as follows : (a) The Loss L l due to Piston-friction. It may be assumed that piston-friction consumes from 10 to 20 per cent of the total available work. (V) The Loss L 2 due to Pipe-friction. The loss of head in the supply-pipe of diameter d^ _ 4//Q) 2 The loss of head in the discharge-pipe of diameter d z 35 2 LOSSES OF ENERGY IN HYDRAULIC ENGINES, ETC. Hence the total loss of head in pipe-friction is The loss in the relatively short working cylinder is very small and may be disregarded. (c) The Loss L 3 due to Inertia. The work expended in moving the water in the supply -pipe _ wA v*_ gn 7' and in moving the water in the discharge-pipe wA v* gn* , 2 ' The total work thus expended , ' nl2g and it may be assumed that nearly the whole of this is wasted. Hence the corresponding loss of head is _ wA_n i'\ f ^_ w_[i_ /'W ^ 3 ~~ A2r\n ' n'l2g " 2r\n ' n'l2g ~ *2g' (d^ The Loss L 4 due to Curves and Elbows. The losses due to curves and elbows may be expressed in the form Z 4 =/ 4 J(Chap. II, Art. 14). (e] The Loss L R due to Sudden Changes of Section. The loss of head in the passage of the water through the ports may o be expressed in the form/' . The loss occasioned by valves may also be expressed by r^ f 27- HYDRAULIC BRAKE. 353 Thus the total loss is V 1 2,2 The coefficient y may be given any desired value between o and oo by turning a valve, so that any excess of pressure may be destroyed and the speed regulated at will. (/") The Loss L 6 due to the Velocity with which the Water leaves the Discharge-pipe. ^ 2g *2g Hence the effective head = ^ - (L^ + L 2 + L 3 + L^+ L 5 + Z 6 ), and the efficiency == I - ^ + L 2 + Z 3 + Z 4 + L 5 + Z 6 ). /'o 6. Brakes. Hydraulic resistances absorb energy which is proportional to the square of the speed. This property has been taken advantage of in the design of hydraulic brakes for arresting the motion of a rapidly moving mass, as a gun or a train, of weight W. In Fig. 200 the fluid is allowed to pass \ H a FIG. 200. from one side of the piston to the other through orifices in the piston. Let m be the ratio of the area of the piston to the effective area of the orifices. Let v be the velocity of the piston when moving under a force P. Let A be the sectional area of the cylinder. 354 HYDRAULIC BRAKE. Then the work done per second = Pv = the kinetic energy produced (m I )V and therefore 9 P= wA(m O 2 , ' 2 g which is the force required to overcome the hydraulic resistance at the speed v. Let V be the initial value of 7\ and 1\ the maximum value of P. Then 1\ = wA(m - i) 2 > 2g Let F be the friction of the slide. Then V* P + F = wA(m i) 2 + F, and P } -f- F is the maximum retarding force. It would cer- tainly be an advantage if t f he retarding force could be constant. In order that this might be the case (m i)?> must be con- stant, and therefore as v diminishes m should increase and consequently the orifice area diminish. Various devices have been adopted to produce this result. Assuming the retarding force to be constant, let x be the piston's distance from the end of the stroke when its velocity is v. Then and therefore T 2 is proportional to x. But (in i )T ; is constant. Therefore (m I ) is inversely proportional to Vx. EXAMPLES. 355 EXAMPLES. 1. A 4-ton hydraulic jack with a 2-in. ram and a i-in. plunger is to lift a weight of i ton, and is worked by a handle with a leverage of 12 to i. If the efficiency of the jack is 80 per cent, what force must be applied to the handle ? Ans. 52^ Ibs. 2. The ram of an hydraulic press has a sectional area- 50 times as great as the pump-plunger. The mechanical advantage of the lever is 10 to i. If a force of 50 Ibs. is exerted on the handle, find the pressure on the ram. Ans. 25,000 Ibs. 3. A force of P Ibs. is required to punch a hole of rtMns. diameter. Firtd the diameter of the ram, the available fluid pressure being p Ibs per square inch. If this pressure is developed by a steam-intensifier with a steam-piston area n times that of the intensifier's ram, find the required steam-pressure. /~^LP $ Ans. \' ; -. up n 4. In a steel hydraulic press the fluid pressure is 6000 Ibs. per square inch, and the maximum allowable stress in the metal is 18,000 Ibs. per square inch. If the internal diameter of the press is 12 ins., what must the thickness of the metal be ? If the thickness of the metal is 3 ins., what must the internal diameter be? Ans. 2.485 ins.; 14.485 ins. 5. A straight-line law is found experimentally to connect the weight W to b'e lifted and the effort E on the handle. Find the law from the following data: when W = 1605 Ibs., E = 10 Ibs., and when W 7 = 6805 Ibs., E = $o Ibs. A pressure-gauge gives the fluid pressure as 1932 Ibs. per square inch, when W = 7000 Ibs. ; find the frictional loss at the leather, and if there is the same percentage of loss at the two leathers find the law connecting E and the force P on the plunger. The experi- ments were made on a jack with a 2^-in. ram, a f-in. plunger, and a lever with a velocity ratio of 30. (Perry.) Ans. W = 305 + 130 E; 9.1 per cent; P = 41 +17.5 E. (Perry's " Applied Mechanics.") 6. An accumulator-ram is 8.8 ins. in diameter and has a stroke of 21 ft. Find the store of energy in foot-pounds when the ram is at the top of its stroke and is loaded till the pressure is 750 Ibs. per square inch. Ans. 958,320 ft. -Ibs. 7. In a differential accumulator the diameters of the spindle are 7 ins. and 5 ins ; the stroke is 10 ft. Find the store of energy when full and loaded to 2000 Ibs. per square inch. Ans. 377,000 ft. -Ibs. 8. The pressure on a 5-in. ram is to be 1000 Ibs. per square inch, and 35 6 EXAMPLES. the supply comes from a tank 100 ft. high. Find the necessary diameter of the piston in the intensifier. Ans. 24 ins. 9. In a differential press the diameters of the upper and lower portions of the ram are 6 ins. and 8 ins. respectively. The pressure is 1000 Ibs. per square inch, and the stroke is 10 ft. Find the load on the accumu- lator, the maximum store of energy, and the store of water. Ans. 22,000 Ibs. ; 220,000 ft. -Ibs. ; i\\ cu. ft. 10. What load must be applied to a differential accumulator to give a pressure of 1600 Ibs. per square inch ? The upper and lower diame- ters of the ram are 3 and 3! ins. respectively, and the friction of the cup- leathers may be taken as 5 per cent of the gross load. Ans. 6062 Ibs. ; 6700 Ibs. 11. Find the weight which will give an average fluid pressure of 750 Ibs. per square inch in an accumulator with a 14-in. ram and a stroke of 16 ft. How much energy can be stored up ? Find the friction at each cup-leather, assuming that between slow rising and falling the pressure fluctuates between 780 and 738 Ibs. per square inch. If tfce pressure is 750 Ibs. per square inch at mid-lift, find the actual fluctua- tion. Ans. 1 15,500 Ibs. ; 1,848,000 ft.-lbs. ; 3234 Ibs. ; 3769 Ibs. 12. An accumulator, loaded to a pressure of 750 Ibs. per square inch, has a ram of 21 ins. diameter, with a stroke of 24 ft. How much H.P. can be obtained for a period of 50 seconds ? Ans. 226.8. 13. An accumulator under a load of 200,000 Ibs. is to transmit 100 H.P. through a4-in. pipe i mile long with a loss of 10 per cent. What should be the diameter of the ram, the coefficient of pipe friction being .006 ? Ans. 17.33 ' 11S - 14. A steam-accumulator has to develop a total force of 66,000 Ibs. upon the ram of a punch. The piston area is 15 times that of the hy- draulic-cylinder, which has a diameter of 10 inches. Find the intensities of the steam and the water-pressure,. Ans. 56 Ibs. ; 840 Ibs. 15. The piston and ram areas of a steam-accumulator are in the ratio of 10 to i. 'Find their diameters so that a steam-pressure of 100 Ibs. per sq. in. may develop a total load on the ram of 38,500 Ibs. Ans. 22.136 ins. ; 7 ins. 16. A Brotherhood engine with a 4-in. cylinder and a 3-in. stroke makes 50 revols. per minute. The average motive pressure is 700 Ibs. per sq. in., and the average back pressure, due to frictional resistances, etc., is 210 Ibs. per sq. inch. Find the H.P. developed, and also deter- mine the diameter of the cylinder if only one half o>i this power is to be developed. Ans. 7 ; 2.83 ins. 17. A crane with an hydraulic efficiency of .9 and a mechanical efficiency of .45 is worked by water at a pressure of 750 Ibs. per sq. inch. The piston has an effective area of 96 sq. ins. on one side, 48 sq. ins. on the other, and pushes a three-sheave pulley-block. Find the maximum weight which can be lifted and the work done per gallon of EXAMPLES. 357 water, first when the water presses on one side only, and second when it presses on both sides. Also find the work done per gallon of water when the full loads in the two kinds of working are being lifted. Ans. 4860 Ibs. ; 6998.4 ft.-lbs.; 2430 Ibs.; 3499.2 ft.-lbs. ; 6998.4 ft.-lbs. 18. An hydraulic crane with a velocity ratio of 9 and a mechanical efficiency of .75 has to lift a weight of 10,000 Ibs. It is worked by water at a pressure of 750 Ibs. per sq. in., and the frictional loss of pressure is 91 Ibs. per sq. inch. Find the diameter of the ram. Ans. 15.2 ins. 19. The two wire ropes from the cage of a ram-lift pass vertically over a pulley to a counterweight, and the ram rises from 100 ft. below to 20 ft. above the level of the supply-pipe. Water-pressures of 500 ibs. and loo Ibs. per sq. in. act upon a 3|-m. and a 7-in. ram, respectively. Find the weight of the ropes per lineal foot and the lifting force at the top and bottom of the stroke. Ans. 4.2 Ibs., 16.7 Ibs. ; 5230 Ibs., 4729 Ibs. ; 55.21 Ibs., 3516 Ibs. 20. Find the pressure due to inertia at the end of the out-stroke of a rotary motor with a 4-m. piston and a 7-in. stroke, driven by water in a 4-111. supply-pipe 250 tt. long. The motor makes 125 revols. per minute, and the length of the connecting-rod is 15 inches. Ans. 20.7 Ibs.; 12.9 Ibs. 21. A direct-acting lift has a ram 9 inches diameter, and works under a constant head of 73 feet, of which 13 per cent is required by ram fric- tion and friction of mechanism. The supply-pipe is 100 feet long and 4 inches diameter. Find the speed of steady motion when raising a load of 1350 Ibs., and also the load it would raise at double that speed. (f = .00672.) If a vaive in the supply-pipe is partially closed so as to increase the coefficient of resistance by 5$, what would the speed be? Ans. Speed = 2 ft. per second ; load = 150 Ibs. 22. Eight cwt. of ore is to be raised from a mine at the rate of 900 feet per minute by a water-pressure engine, which has four single-acting cylinders, 6 inches diameter, 18 inches stroke, making 60 revolutions per minute. Find the diameter of a supply-pipe 230 feet long for a head of 230 feet, disregarding resistances and taking/ = .006. Ans. Diameter = 4 inches. 23. If A. be the length equivalent to the inertia of a water-pressure engine, F the coefficient of hydraulic resistance, both reduced to the ram, z/ the speed of steady motion, find the velocity of ram after moving from rest through a space x against a constant useful resistance. Also find the time occupied. 24. An hydraulic motor is driven from an accumulator, the pressure 35 8 EXAMPLES. in which is 750 Ibs. per square inch, by means of a supply-pipe 900 feet long, 4 inches diameter; what would be the maximum power theoreti- cally attainable, and what would be the velocity in the pipe correspond- ing to that power ? Find approximately the efficiency of transmission at half power,/ = .007. Ans. H.P. = 250; v = 22 ft.; efficiency = .66 nearly. 25. A gun recoils with a maximum velocity of 10 feet per second. The area of the orifices in the compressor, after allowing for contraction, may be taken as one twentieth the area of the piston. Find the initial pressure in the compressor in feet of liquid. Assuming the weight of the gun to be 12 tons, friction of slide 3 tons, diameter of compressor 6 inches, fluid in compressor water, find the recoil. Find the mean resistance to recoil. Compare the maximum and mean resistances, each exclusive of friction of slide. Ans. 621 ; 4 ft. 2$ in. ; total mean resistance = 4.4 tons; ratio = 2.5. CHAPTER V. IMPACT, REACTION, IMPACT AND TANGENTIAL TURBINES. NOTE. The following symbols are used: v l = the velocity of the jet before impact ; i> z ' ' " i . i i a fter leaving the vane ; u = " " " " vane; F= " " " " water relatively to the vane; A == sectional area of the impinging jet ; m = mass of the water reaching the vane per second. i. Impact of a Jet upon a Flat Vane Oblique to the Direction of the Jet. Let 6 be the angle between the normal to the vane and the direction of the impinging jet, the angle between the normal to the vane and the direc- tion of the vane's mo- tion, and a the angle between the vane and the vertical. The jet, moving with its stream-lines parallel, swells out near the vane, over which it spreads and with which it travels along in the direction of the vane's motion, and finally again flows along with its stream-lines sensibly parallel to the vane. 359 360 IMPACT ON FLAT YANES. The problem is still further complicated by the production of eddies and vortices for which allowance can only be made in a purely empirical manner. Let N be the normal pressure on the vane due to the impact. Let N' be the total normal pressure on the vane. Let J'Fbe the weight of water on the^vane. Then N N' - W sin a = change of momentum in direction of the normal = mi\ cos mu cos 0, or N = m(v l cos 6 u cos 0), ... (i) (N.B. The sign in front of u cos will be plus if the jet and vane move in opposite directions.) The term W sin a maybe designated the static pressure, and the term m(v l cos 6 u cos 0) the dynamic pressure, which causes the deviation of the stream-lines. NOTE. The pressure when a jet_/?r^/ strikes the plane is greater than when the flow has become steady, or a permanent regime is established. This is made evident by the following consideration: At any moment let MN, PQ, *RS be the bounding planes across which the water is flowing with its stream-lines sensibly parallel. In a unit of time let the bounding planes of the mass be M'N', P'Q', R'S'. Then, initially, the reaction of the plane must destroy the motion of the mass of the fluid bounded by M'N' , P 'Q ', and R'S'. Take OC to represent i\ in direction and magnitude. In one second the vane AB moves parallel to itself into the position A'B'. Let A'B' intersect OC in D. IMPACT ON FLAT k,1NES. 301 Then m = -A . DC = -A(v, - OD) g g ' W I COS 0\ = -A(v. u ~) ....... (2) \ l cos Qi . g \ l cos Q Thus equation (i) becomes w A N = - z(v, cos 6 - u cos 0) 2 . ... (3) g cos 6^ Let P be the pressure in the direction of the vane's motion. Then w cos /> = A^ cos = -A ^- (v, cos - u cos 0) 2 , . (4) and the useful work done on the vane per second A g cos = Pu = A --- ,-,#(7', cos u cos 0) 2 . (5) cos W V 3 The total available work = -A ..... (6) g 2 W COS ^4 - ^,2/(>i COS G U COS 0) 2 LJ 4.u ^r J?" cos ^ Hence the efficiency =. e __ COS W = 2 C^ST^ 3^1 C S This is a maximum when v l cos ^ = 3// cos 0, ..... (8) and therefore o the maximum efficiency = cos 2 6. . . (9) If the vane is of small sectional area, a portion of the water will escape over the boundary and the pressure must necessarily be less than that given by equation (3). 362 IMPACT ON FLAT YANES. Series of Vanes. Instead of one vane moving before the jet, let a series of vanes be introduced at short intervals at the same point in the path of the jet. The quantity of water now reaching the vane per second is evidently w m=-Av l9 (10) o and, by equation (i), the normal pressure w N =. ~Av l (v l cos V u cos 0). . . . (11) o Also, the pressure in the direction of the motion of the vane = P A^cos = Av l (v l cos 6 u cos 0) cos 0. (12) FIG. 202. l the maximum efficiency^ (20) Again, if u = o, i.e., if the vane be fixed, and if H be the head corre- sponding to the velocity 7/1, then, by equation (17), P Av? 2wAH g = twice the weight of a column of water of height //and sectional area A. Ex. 2. Let each of a series of vanes be at right angles to, and move in the line of, the jet's motion at the instant of impact. Then 6 = o = 0. Hence the pressure N = P = Av*(vi u); . . . . . (21) the useful work = Pu = Av\u(y\ u)\ (22) & , a- 2u(Viu] the efficiency = ; ; (23) the maximum efficiency = (24) Ex.3. A stream of .125 sq. ft. sectional area delivers locu. ft. of water per second and impinges normally against a flat vane. It is required to find (a) the pressure on the vane if fixed ; (ff) the pressure and the useful effect if the vane moves in the direction of the jet's motion with a velocity of 40 ft. per second ; ( 90, and the coefficient /? 2 sin 2 -, or I cos /3, is greater than unity. If the surface be of the cup type and hemispherical, the 1 80 maximum efficiency = sin 2 = I, since j3 = 180. The water should therefore leave the surface without velocity, and, substituting ^ = 2u and ft = 180 in equation (i), v = M 2 _[_ (2u lif 2U(2U 11} = O. Ex. A jet of water of .125 sq. ft. sectional area delivers 12 cu. ft. of water and impinges axially upon a 120 cone. Find (a) the pressure on the cone when fixed, and (b) the pressure on the cone and the useful 368 IMPACT ON BORDERED YANE. effect when the cone is driven in the direction of its axis with a velocity of 32 ft. per second. The velocity of the jet before impact = -^ = 96 ft. per sec. 62i , 60 (a) Pressure on convex surface = 2 ^ . 12.96 sm a = 1125 IDS. Pressure on concave surface = 212.96 sin 2 - = 3375 Ibs. (b) When the water impinges on the convex surface the work done = 2 --^32(96 32)" sin 2 = 16,000 ft.-lbs., 16000 the pressure = = 500 Ibs. *) When the water impinges on the concave surface the work done = 2 --32(96 32) 2 sin 2 ^- = 48,000 ft.-lbs., 32 5 2 A8000 the pressure = = 1500 Ibs. 6. Impact of a Jet of Water upon a Vane with Borders. Let the vane in Art. I be provided with borders, Figs. 207 and 208, so as to produce a further deviation of the stream- lines, and let the water finally flow off with a velocity v 2 in a direction making an angle 0' with the normal to the vane. FIG. 207. FIG. 208. Then the normal pressure = N = mv l cos =F mv^ cos 0' ^p mu cos m(v t cos T ?' 2 cos 6' ^f u cos 0), the sign of the second term being plus or minus according to the direction in which the stream-lines are finally deviated. IMPACT APPARATUS. 3 6 9 The effect of the borders is therefore to increase or diminish tfre normal pressure, and hence also the useful work and the efficiency. SPECIAL CASE. Let the vane be at rest, i.e., let u = o, and let the final and initial directions of the jet be parallel. Also, let v l = v r Then N = m(v l cos 6 + v l cos 6) w = 2 AV? cos e g cos 6. Hence, if 6 = 0, the normal pressure N ^.wAH four times the weight of a column of water of height H and sec- tional area A. 7. Impact Apparatus in Hydraulic Laboratory, McGill University. This apparatus was constructed for the purpose of determining the force with which jets from orifices, nozzles, etc., impinge upon vanes of different forms and sizes. A massive cast-iron bracket, Fig. 209, has one end securely bolted to the front of the tank, and the other supported by a vertical tie-rod from one of the oak beams in the ceiling. The upper surface is provided with accurately planed slides, which are set level about 5 ft. above the orifice axis. If, from any cause, the end of the bracket farthest from the tank is found to be too high or too low, the error can be corrected by loosen- ing or tightening the nut on the tie-rod. The balance proper is carried by a sliding frame which can be moved horizontally into any position along the bracket by means of a rack and pinion actuated by a sprocket-wheel with chain. At one end the frame has two equal arms with a common horizontal axis parallel to the bracket, and each arm has a stop on its lower surface which serves to limit the oscil- lation of the balance. 37 IMPACT APPARATUS. The balance, in its mean position, consists of a main trunk with horizontal axis rigidly connected with a vertical slotted arm and with two equal horizontal arms at one end. The common axis of the latter is horizontal and perpendicular to the axis of the main trunk. The hardened-steel knife-edges of the balance are 4 ft. centre to centre and rest in hardened- steel vees inserted in the ends of the sliding frame on each side of the bracket. The bottom of each vee is in the same FIG. 209. horizontal line (called the axis of the vees) at right angles to the bracket. A bar with the upper portion graduated in inches and tenths has a slot in the lower portion, which is bent into a circular segment of 9^ ins. radius. The bar slides along the slot in the vertical arm of the balance. A radial block, with the holder into which the several vanes are screwed, moves along the slot in the circular segment, and may be clamped in any required position, the angular deviations from the vertical COEFFICIENT OF IMPACT. 37 1 being shown by graduations on the segment. The centre of this segment in every case coincides with the central point of impact on a vane, is in the vertical axis of the balance-arm, and is also vertically below the axis of the vees. Thus the jet can always be made to strike the vane both centrally and normally. The scale-pan hangs from a knife-edge at one end of the horizontal arms of the balance, while to the other end is attached a fine pointer, which indicates the angular movement of the balance on a graduated arc fixed to the sliding frame. The balance is in its mid-position when the pointer is opposite the zero mark. When a vane has been secured in any given position, the preliminary adjustment of the balance is effected by moving a heavy cast-iron disc along a horizontal screw fixed into the main trunk. The sensitiveness of the balance is also increased or diminished by raising or lowering heavy weights on two vertical screws in the top of the trunk. Assume that the adjustments have all been made and that the jet, Fig. 210, now impinges normally upon a vane. Let Wbe the weight required in the scale-pan to bring the balance back into its mid-position. Let F a be the actual force of im- pact determined by the balance. Let F t be the theoretical force of impact deduced by the ordinary formulae. Fa Then the ratio = = c may be called the coefficient of r t impact. Let y be the vertical distance of the central point of impact below the horizontal axis of the orifice, which is 36 ins. below w 372 COEFFICIENT OF IMPACT. the axis of the vees. The distance between this axis and the point of suspension of the scale-pan is 24 ins. Let v be the velocity with which the water issues from the orifice. Let v' be the velocity of the jet at the point of impact. Then Q being the delivery per second and ft the angle through which the water is turned on the vane. If the axis of the jet at the point of impact makes an angle 6 with the horizontal, then v' cos 6 = v = c v V2gh. Therefore tv /3 F t cos = 2~~Qv sin 2 . Again, taking moments about D, F a cos #(36+7) = W. 24. Hence F ft 6W ft' wc ~ O O " W V -'--, ...... (5) and w -Qu(y^ - u] the efficiency 2u IV V* U Z ^ + U g 2~" Thus the more nearly v^ is equal to u y and therefore the larger the area A of the orifice, the greater is the efficiency. If the vessel is driven in the same direction as the jet, then v l -f- u is the relative velocity of the jet with respect to the earth, and the reaction is R = horizontal change of momentum w w = Q(VI + u ~) =~< : s v Av l (v l + ^ <~> <5 w = j^i(*i + ). ........ (7) disregarding the contraction and putting c v = I. SCOTCH TURBINE. 375 9. The Jet Reaction Wheel (Scotch Turbine). In this fejni of motor the water enters the centre of the wheel, spreads out radially in tubular passages, and issues from openings at the ends tangentially to the direction of rotation. FIG. 212. FIG. 214. FIG. 213. FIG. 215. Fig. 2 1 2 represents the simplest wheel of this class. In England it is known as Barker's Mill, and in Germany as Segner's Water-wheel. A reaction wheel may have several tubular passages as in Fig. 214, while the vertical chamber XY may be cylindrical, prismatic, or conical. The Scotch -or Whitelaw's turbine, Fig. 215, does not differ essentially, excepting in the curved arms, from the simple reaction wheel. Let r be the horizontal distance between axis of orifice and axis of rotation. " h lt " head of water over the orifkes when closed, 37 6 SCOTCH TURBINE. Let V be the velocity of efflux relatively to the tube when the orifices are open. " u " " corresponding linear velocity of rotation at the centre of an orifice. " 7' 2 " " absolute velocity of efflux = V u. " Q " " discharge. " R " " reaction. Then P. = */( + V*), ..... (I) c v being the coefficient of discharge. Also, wQ (V 11) horizontal linear change of momentum o = reaction producing rotation = R (2) The useful work = Ru= r (ru)u. (3) (4) The efficiency Ru_ _ (V u)u _ 2(V ^ u)u ~ wQh ~ .gh ~T~ 2 ~ ' -7~2 M c v Again, the efficiency _ (V - u)u _ M* IV \ gh ~ gh\u I * lf T , ^-_ - gh\ l " ' u* I = j j ^Jl+ ~T "" terms containing higher powers of j I [ . Thus the efficiency must theoretically increase with u, but the value of n is limited by the practical consideration that, even at moderately high speeds, so much of the head is SCOTCH TURBINE. 377 absorbed by frictional resistance as to sensibly diminish the efficiency. The serious defects of the reaction wheel are that its speed is most unstable and that it admits of no efficient system of regulation for a varying supply of water. By equation (4), the efficiency is a maximum, for a given value of u, when F 2 - 2 Vu + c v V = o, or F=(i + V - 2 = o = F 9 n, then = u v -- 2in\ cos y and 2/' COS 7/ 2 2// = 2g(H or UV t COS y -.= gH T , an equation giving the best speed of the turbine. H is the head required to give the velocity v^ at entrance. H l is the total head under which the turbine works. There should be no loss in shock at entrance, and to insure this ad ( Fj), the relative velocity, must be tangential to the lip at a. The lip angle a is then given by u sin (a + y) - = -- - - cos y 4- cot a sin y, 7 ' 7 3^4 BORON'S TURBINE. or u cot a cosec y cot y. v \ Since u = V 2 , the triangle fgh is isosceles, and e v 9 221* sin . 2 2 2 wQ i v *^\ The useful work = 77(1 ), if being the efficiency. Let R be the mean radius. " / " " water thickness, measured radially. Then, sin c = Q. Allowance may be made for the principal hydraulic resist- ances (friction, etc.) by taking f 2 t-L to represent the loss of head up to the inlet, and f. ?- " '* 4< " <4 " in the wheel-passages. *g Then and '2' / 2 and/ 4 being coefficients to be determined by experiment. Usually / 2 varies from .025 to .2 and upwards, an average value being .125, and/ 4 varies from .1 to .2. The normal distance, Fig. 221, between two consecutive vanes should be > the stream's normal thickness between the BURDIN'S WHEELS. 385 vanes, i.e., > v? X the normal thickness of the stream before m impact. FIG. 221. FIG. 222. Burdin's (Fig. 222) is among the best of impact wheels, differing only from the simple Borda in receiving the water at several points simultaneously and in distributing the outlet openings in three concentric circles. Ex. A 5-H.P. Borda turbine, of 4 ft. mean diameter and 4 ft. depth, works under a total head of 20 ft. The direction of the jet before impact is inclined at 33 33' (y} to the horizon, and the angle of exit (e) is 19 8'. The jet delivers 3 cu. ft. of water per second. Find (a) the best speed of the turbine; (b) the lip angle a; (c) the velocity, 2/ a , of the water as it leaves the turbine; (d) the hydraulic efficiency; (e) the practical effi- ciency. 20 4 = 16 head required to produce t/j = . (a} Therefore v,. = 32 ft. per second. The best speed is then given by uv\ cos Y = r ffi> or . 3 2 cos 33 33' = 3 2 - 20 or u . 32 = 32.20 . and u = 24 ft. per second. The number of revolutions per minute = ' sin Z) == ^. = fi == i = 2/1 32 4 cos y -f cot ct sin or and cot (i 80 - a) = cot 33 33' cosec 33 33' = .1509, a = 98 35'. 3 86 DANAIDES. (c) Assuming F a = u, then z/a = 2 sin = 48 sin 9 34' = 48 x ^ = 8 ft. per second. (d) The hydraulic efficiency = i = ~ = -9S- 64. 20 20 (e) If 77 is the practical efficiency, 7.621.3(20-5-^) = 5.550 and rj = .772. Danaides. These are wheels capable of revolving about a vertical axis and consist of two casings which are more or less in the form of inverted truncated cones (Fig. 223) and which enclose a space divided into a number of water-passages by vanes which may be flat, spiral, or screw-shaped. In the wheels described by Belidor the inner casing with the vanes FIG. 223. FIG. 224. attached is made to closely fit the outer conical casing, which is fixed. In another form of Danaide the vessel is divided into two equal parts by a vertical partition. Thus in wheels of this type the water approaches the axis in its descent, developing a centrifugal force which must be taken into account. Consider the case of a Danaide with double conical casing and flat vertical vanes, Fig. 224. DANAIDES. 3*7 The relative velocities V l , F 2 are evidently at right angles to , the corresponding peripheral velocities at inlet and exit. A Therefore v* = V? + u? and v*= F 2 2 + u*. Also, if h 2 is the depth of the wheel, 772 772 u 2_ u z V 2 _ * 1 I _ ffl _ ?!. " " 2,f 2g u _ the term 2 L being due to the effect of centrifugal force. <> Hence and the mechanical effect Tzib-wheel. This form of impact wheel, Fig. 225, consists of a number of floats fixed to a vertical shaft. The wheel is either fitted into a well, a small clearance being allowed, or it is given a larger diameter and is placed just below the well. The water is brought along a properly designed race, enters the well tangentially with considerable velocity and acquires a rotary motion. Thus it acts upon the floats both by impact and by pressure. The efficiency of the wheel is small, as a large portion of the water escapes without producing its full effect. Practical experience indicates that the best speed of the middle of the floats is about one third of the velocity of the current, and that the efficiency varies from 15 to 40 per cent, but rarely exceeds 30 per cent. FIG. 225. IMPACT ON CURBED l/ANES. ii. Jet impinging upon a Curved Vane and deviated wholly in one Direction Best Form of Vane. Let the jet, of sectional area A, moving in the T\ direction AB with a velocity v^, v\ \ drive the vane AD in the direction AC with a velocity u, Fig. 226. \aj~~ | V x ^ Take A B to represent v^ in direc- \ / ^~p tion and magnitude. \ / Take AC to represent u in direc- \ / ^ tion and magnitude. \ J in CB - "V"-""" ~^ B Then CB evidently represents ^C / ^ ^e velocity of the water relatively L to the vane, in direction and magni- FIG. 226. tude. If CB is parallel to the tan- gent to the vane at A, there will be no sudden change in the direction of the water as it strikes the vane, and, disregarding friction, the water will flow along the vane from A to D with- out any change in the magnitude of the relative velocity V (= CB). The vane is then said to " receive the water with- out shock." Again, from the triangle ABC, denoting the angles BAC Y ABC, ACB, by A, B, C, respectively. u __ A C _ sin B _ sin B ^ ~ ~AB ~ sin C ~ sin (A + B) J ' ' ' and therefore v cot B = - cosec A cot A, . . . '. (2) a formula giving the angle, between the lip and the direction of the impinging jet, which will insure the water being received " without shock." In the direction of the tangent to the vane at Z>, take DE = CB (= V}. IMPACT ON CURBED VAXES. 389 Draw DF parallel and equal to AC (= u). , Complete the parallelogram EF. Then the diagonal DG evidently represents in direction and magnitude the absolute velocity ?> 2 with which the water leaves the vane. Draw AK equal and parallel to DG (= v 2 ). Join BK. Then BK represents the total change of velocity between A and D in direction and magnitude. Thus if R is the resultant pressure on the vane, then R =m. BK. Let ML be the projection of BK upon AC. Then ML represents the total change of velocity in the direction of the vane's motion. Let P be the pressure upon the vane in this direction. Then P = m . LM. ...... (3) v 2 _ v 2 The useful work Pu = mil . LM '= m -. . 4 W V The total available work = A- 1 -. g 2 mu . LM v? v? The efficiency ^-^ = *T^r ..... (6) g 2 Again, join CK. Then, since AC is equal and parallel to DF, and AK to DG, the line CK is equal and parallel to DE, and is therefore equal to CB. Thus in the isosceles triangle CBK, CB is equal and parallel to the relative velocity V at A , CK is equal ' and parallel to the relative velocity V at D, and the base BK repre- sents the total change of motion. ?9 IMPACT ON CURBED YANES. Let d be the angle through which the direction of the water is deviated, i.e., the angle between AB and AK. Then F 2 = CK^ AK* + AC* 2AK .AC cos (A + 6) V% + U 2 2V 2 U COS (A -h bisects the angle between BC and BH, and this angle is equal to the angle between the tangents to the vane at A and D. Let a be the angle between the normals at A and D. Then the angle KCB = a, and the angle CBK= -(180 - a) = 90 - -. Therefore BK = 2<^(cos 90 - ?) = 2 F sin ^. Hence R = m . BK = 2mV s'm . . . . (10) Let X, Y be the components of R in the direction of the normal at A and at right angles to this direction. Then X = R cos - ;Fsin a, ....... (ll) ex. a Y R sin 2mVs'm 2 - = mV(\ cos a). . (12) 2 2 IMPACT ON CURVED VANES. 39 * The efficiency is a maximum when d(Pu\ dP , -A = o = u- r -\-P. .... (13) du du The efficiency is nil when Pu = o, i.e., when u = o or P = o. . . (14) In the latter case, since P m . LM, the projection must be nil, and therefore FIG. 227. BK must be at right angles to AC, as in Fig. 227. The angle ^4 (7.5 is now = 1 80 -. Therefore u _ sin ABC v, ~ sin ACB or sin - 2 If ^7T is parallel to ", Fig. 228, then the angle FIG. 228. and therefore sn v l sin ACB COS 39 2 IMPACT ON CURBED YANES. Let the direction of the impinging jet be tangential to the vane at A, Fig. 226, and let the jet and vane move in the same direction. Then V =. v l u, m = A(v l u) ; <5 PY A(y l u)\i cos a) = 2~A(i\ ) 2 sin 2 -; <5 O useful work = Pu = 2 Au(v l u) 2 sin 2 -; rr . efficiency 4 This is a maximum and equal to sin 2 when i\ ^u. These results are identical with those for a concave cup when a 180. Instead of one vane let a series of vanes be successively introduced at short intervals at the same point in the path of the jet. Then w m=-.Av l9 and hence the pressure P, useful work, and efficiency respec- tively become w A w v? v* v* v* Av,.LM\ Av -i; - 1 ^^. g g 2 v? N.B. Frictional resistance may be taken into account by assuming that it absorbs a fractional portion of the head corre- sponding to the velocity of the jet relatively to the surface over F 2 which it spreads. Thus the loss of head =/ , and the o J/ 2 corresponding loss of energy = wQf . EXAMPLE. 393 Ex. A curved vane in the form of the quadrant of a circle 'without shock, at an edge, a stream of water flowing at the rate p&r second, which drives the vane with a velocity of 4 ft. per second in a direc- tion making an angle of 60 with the re- ceiving edge. At the receiving edge the triangle abc is a triangle of velocities in which the angle abc \20,ac= 12 ft. ,# = 4 ft., and be = V, the relative velocity at a which must be parallel to the tangent at a t as there is to be no loss in shock. Then receives of 12 ft. or and 1 2' = 4" + V 2 2.4 V cos 1 20, 128 = V + 4^, V= 9.4891 ft. per second. Also, if y is the angle between ac and the receiving edge, then the angle cab = 60 y, and 9.4891 _ be _ sin (60 y} . or cot y = 3.3166 and y = 16 47'. At the discharging edge fghk is the parallelogram of velocities in which fg, parallel to ab, = 4 ft., fk, tangental at /, = 9.4891 ft., the relative velocity, and/^ is the absolute velocity in direction and magni- tude with which the water leaves the vane. Let the angle hfk = S. Then TV = 4 2 + (9-489I) 2 2 x 4 x 9.4891 cos 30 = 40.2993, and Vi = 6.3481 ft. per sec. 891 fk sin (d + 30) Again, and . = cos 30" + s,n 30" cot 5, cot 8 3.0126, or 8 =. 18 22'. 12. Tangential or Centrifugal Turbines. Suppose that the vane AD is constrained to revolve about a vertical axis O with a constant angular velocity GO. If OP, OQ are consecu- tive radii and if PN is drawn at right angles to OQ, then the 394 TANGENTIAL TURBINES. work of the centrifugal force as a mass m of fluid moves from P to the consecutive position Q = ma?r . QN = mo&r . dr, where OP = r. FIG. 230. The total work in the movement from A to D r* = I mo&r dr i: Uj , U 2 being the linear velocities at A and D respectively. u. a u 2 . If the flow is from A to D, - is evidently a gain of 2 g head, while it is a loss of head if the flow is from D to A. In tangential or centrifugal flow turbines a number of vanes are encased and have concentric inlet and outlet surfaces. The flow, which is more or less radial, is towards the axis in the inward-flow and from the axis in the outward-flow turbine. Since the axis of rotation is vertical, the effect of gravity may be disregarded. TANGENTIAL TURBINES. 395 If v r ' t v r " are the radial components of v l and v 2 respec- jtively, v r ' v l sin y and v r " F 2 sin ft. FIG. 231. FIG. 232. Then, by the condition of continuity of flow, and dis- regarding the thickness of the vanes, 2nr^d^Ur = ^7^X1^1 sin y Q = and Q r " 2nr. 2 d 2 V 2 sin ft, sin A ... (i) and ^ 2 being the inlet and outlet depths of the wheel. First. Disregard hydraulic resistances. Then y 2 _ Y 2 I 2 _ 2 r 2 ~ F T ~f- 2 j . . (2) cos -* = v* 2u^ cos y 4. ^ 2 2 , or COS . (3) 396 TANGENTIAL TURBINES. Also, from the triangle *,'= -2 J>, cos / ...... (4) Hence wQ v 2 v 2 the useful work = - - - - wQ = ("i v i cos Y + a F 2 cos ~ 2 2 )- (5) total loss due to the resistances in question Eq. (2) now becomes ( l +/4)^r *i + U 2 u \ > (15) and if H is the head over the inlet, \ = H. ........ (16) 39 8 EXAMPLES. Ex. i. A centrifugal inward flow turbine, with equal inlet and outlet depths and working under the head of 200 ft., passes i cu. ft. of water per second. The angle y is 15; w\ = $r?. ; and it is assumed that u-i. = J/i. Find (a) the peripheral speed; (b) the lip angle at outlet; (c) the energy carried away by the water; (d) the energy lost in hydraulic resistance; (e) the useful work ; (/) the efficiency. (Disregard the thick- ness of the vanes.) sin 15 .259, cos 15 .966, and let /*=/* = .125. / i\z/i a () * + Q]Z~ = 20 - Therefore z/i = io6f ft. per sec. By eq. (14), V? + u-, i.e., /? must not exceed 45. \2 Second, taking the hydraulic resistances into account, V + v? ='- 17 = (' +/<) V} - u? + u Also, the loss of head up to inlet =f 2 -- m V* f " " " " in wheel-passages =/ 4 ^ = and the total loss of head due to the principal hydraulic resist- ances If // is the head over the inlet, o +/*)-- = H. NOTE. Impact, centrifugal, and jet turbines will work with the axis inclined at any angle to the vertical. 402 RESISTANCE TO MOTION OF SOLIDS IN FLUIDS. 14. Resistance to the Motion of Solids in a Fluid Mass. The preceding results indicate that the pressure due to the impact of a jet upon a surface may be expressed in the form F 2 P= KwA, ig A being the sectional area of the jet, V the velocity of the jet relatively to the surface, and K a coefficient depending on the position and form of the surface. Again, the normal pressure (^V) on each side of a thin plate, completely submerged in an indefinitely large mass of still water, is the same. If the plate is made to move hori- zontally with a velocity F, a forward momentum is developed, in the water immediately in front of the plate, while the plate tends to leave behind the water at the back. A portion of the water carried on by the plate escapes laterally at the edges and is absorbed in the neighboring mass, while the region it originally occupied is filled up with other particles of water. Thus the normal pressure A r , in front of the plate, is increased by an amount ;/, while at the back eddies and vortices are produced, and the normal pressure N at the back is diminished by an amount ;/'. The total resultant normal pressure, or the normal resistance to motion, is n -f- ', and this increases with the speed. In fact, as the speed increases, n approximates more and more closely to N, and in the limit the pressure at the back would be nil, so that a vacuum might be maintained. Confining the attention to a plate moving in a direction normal to its surface, the resistance is of the same character as if the plate is imagined to be at rest and the fluid moving in the opposite direction with a velocity V. So, if both the water and the plate are in motion, imagine that a velocity equal and opposite to that of the water is impressed upon every particle of the plate and of the water. The resistance is then of the same character as that of a plate moving in still water, the velocity of the plate being ,the velocity^relatively to the RESISTANCE TO MOTION OF SOLIDS IN FLUIDS. 43 Water. Thus, in general, the resistance to the motion of such a -plane moving in the direction of the normal to its surface, with a velocity V relatively to the water, may be expressed in the form F 2 R = KwA , A being the area of the plate, and K a coefficient depending upon the form of the plate and also upon the relative sectional areas of the plate and of the water in which it is submerged. According to the experiments of Dubuat, Morin, Piobert, Didion, Mariotte, and Thibault, the value of K may be taken at 1.3 for a plate moving in still water, and at 1.8 for a current moving on a fixed plate. Unwin points out the unlikelihood of such a difference between the two values, and suggests that it might possibly be due to errors of measurement. Again, reasoning from analogy, the resistance to the motion of a solid body in a mass of water, whether the body is v.'holly or only partially immersed, has been expressed by the formula F 2 R KwA , V V being the relative velocity of the body and water, A the greatest sectional area of the immersed portion of the body at right angles to the direction of motion, and K a coefficient depending upon the form of the body, its position, the relative sectional areas of the body and of the mass of water in which it is immersed, and also upon the surface wave-motion. The following values have been given for K\ K i.i for a prism with plane ends and a length from three to six times the least transverse dimension ; K i .o for a prism, plane in front, but tapering towards the stern, the curvature of the surface changing gradually 404 PRESSURE ON PLATE IN PIPE. so that the stream-lines can flow past without any pro- duction of eddy motion, etc. ; JC =. .5 for a prism with tapering stern and a cut-water or semicircular pro\v; K .33 for a prism with a tapering stern and a prow with a plane front inclined at 30 to the horizon ; !' = .16 for a well-formed ship. Froude's experiments, however, show that the resistance to the motion of a ship, or of a body tapering in front and in the rear, so that there is no abrupt change of curvature leading to the production of an eddy motion, is almost entirely due to skin-friction (see Art. i, Chap. II). 15. Pressure of a Steady Stream in a Uniform Pipe against a Thin "Plate AB Normal to the Direction of Motion. The stream-lines in front of the plate are deviated and a contraction is formed at Cf y They then converge, leaving a mass of eddies behind the plate. Consider the mass bounded by the transverse planes C l C l > Cf z ^ where the stream-lines are again parallel. At C l C l let / lf A iy 7'j, 2 l be the mean intensity of the c t r pressure, the sectional area of the waterway, the velocity of flow, and the elevation of the C. of G. of the section above datum. Let / 2 , A 2 , 7' 2 , 2 2 be corre- sponding symbols at Cf. Let / 3 , A lt 7' 1? #3 be corresponding symbols at Cf y Let a be the area of the plate. Let c c be the coefficient of contraction. Neglect the skin and fluid friction between C l C l and Cf y Then, by Bernoulli's theorem, . A , 'i = = 8 , A , !* = : g , A , *>? , (v, - ^'.) 3 W 2g *' 2 W 2g "* U' 2g 2g * PRESSURE ON PLATE IN PIPE. 405 (TV 7; )2 the term -' *'- representing the loss of -head due to the. bending of the stream-lines between C 2 C. t and Cf . Hence Again, let R be the total pressure on the plane. Then , - p^A, = ( A - /3 ) A = j fluid P ressure n the direction ( of the axis, Thus = component of the weight in the direction of the axis. i + wAfa ^ 3 ) R change of motion in direc- tion of axis = o, since the motion is steady. Hence R _ A ~ A But ^^ =3 ^ 2? , 2 ^'^(^ - )z/ 2 . Therefore < (c e (m i) 406 PRESSURE ON CYLINDER IN PIPE. where m 1 , or a R = where K m \ -- r I ! . \c c (m - i) J 16. Pressure of a Steady Stream in a Uniform Pipe on a Cylindrical Body about Three Diameters in Length. The stream-lines in front of the body are deviated and a contraction is formed at C 2 C 9 . They then converge, flow in parallel lines, and converge a second time at C.. 6 , leaving a mass of eddies behind the body. Consider the mass bounded by the planes C l C l , C \C \. As in the previous article, let p l , A lt "c\ , s^ be the intensity of pressure, sectional area of the waterway, velocity of flow, and elevation of C. of G. above datum at G C, Cf r p,, A,, 7- ^r, be similar symbols jr 2, ' i * z J for C\C Z . * / 3 , ^4 3 , ^ 3 , -s- 3 be similar symbols FIG. 235. for Cf v Pi , A lt 7' t , ^ 4 be similar symbols for C" 4 6* 4 . Neglect the skin and fluid friction between C l C l and C \C 4 . Then, by Bernouilli's theorem, Z * ^- being the loss of head between CjC, and C 3 ^ 3 and o / _ , \2 being the loss of head between Cf z and ^ 4 " 4 . O PRESSURE ON CYLINDER IN PIPE. 47 Hence , & \ ~~~ & 4 ~\ A - A _ fe - * w But A l v l = A 2 v z = A^ , c c (A l -a) =A 2 , "~ ' ' and A s = A l a. .. ; " ' '. Therefore m -\ -i) ~ ^P"i j j where m = L . Also, as in the preceding article, Hence = ^ K = m where m L , and i \(m~- i)^ (m i This value of K is always less than the value of K for the plate in the preceding article for the same values of m, a, and c c . Hence the pressure on the cylinder is also less than the corresponding pressure on the plate. In every qase K should be determined by experiment. 48 EXAMPLES. EXAMPLES. 1. A stream with ^ transverse section of 24 sq. ins. delivers 10 cu. ft. of water per second against a flat vane in a normal direction. Find the pressure on the vane. Ans. 1171^ Ibs. 2. If the vane in example i moves in the same direction as the im- pinging jet with a velocity of 24 ft. per second, find (a) the pressure on the vane ; (b} the useful work done; (c) the efficiency. Ans. (a) 42 1 Ibs.; (b) 10,125 ft.-lbs. ; (c] .288. 3. What must be the speed of the vane in example 2, so that the efficiency of the arrangement may be a maximum ? Find the maximum efficiency. Ans. 20 ft. per sec.; / T . 4. Find (a) the pressure, (b) the useful work done, (c} the efficiency, when, instead of the single vane in example 2, a series of vanes are introduced at the same point in the path of the jet at short intervals. Ans. (a) 703i Ibs.; (b} 16,875 ft.-lbs. ; (c) .48. What must be the speed of the vane to give a maximum efficiency ? What will be the maximum efficiency? Ans. 30 ft. per sec.; .5. 5. A stream of water delivers 7500 gallons per minute at a velocity of 15 ft. per second and strikes an indefinite plane. Find the normal pres- sure on the vane when the stream strikes the vane (a) normally; (b) at an angle of 60 to the normal. Ans. (a) 585 9 Ibs.; 292.9 Ibs. 6. A railway truck, full of water, moving at the rate of 10 miles an hour, is retarded by a jet flowing freely from an orifice 2 ins. square in the front, 2 ft. below the surface. Find the retarding force. Ans. 7.97 Ibs. 7. A jet of water of 48 sq. ins. sectional area delivers 100 gallons per second against an indefinite plane inclined at 30 to the direction of the jet ; find the total pressure on the plane, neglecting friction. How will the result be affected by friction ? Ans. 750 Ibs. 8. If the plane in example 7 move at the rate of 24 ft. per second in a direction inclined at 60 to the normal to the plane, find the useful work done and the efficiency. Ans. 2250 ft.-lbs.; ^. At what angle should the jet strike the plane so that the efficiency might be a maximum ? Find the maximum efficiency. Ans. sin" 1 f ; \. 9. A stream of 32 sq. ins. sectional area delivers 7^ cu. ft. of water per second. At short intervals a series of flat vanes are introduced at the same point in the path of the stream. At the instant of impact the direction of the jet is at right angles to the vane, and the vane itself moves in a direction inclined at 45 to the normal to the vane. Find EXAMPLES. 409 the speed of the vane which will make the efficiency a maximum. Also find the maximum efficiency and the useful work done. Ans. 15.08 ft. per. sec.; -/ r ; 2io6||f ft.-lbs. 10. A stream of water of sq. ft. sectional area delivers 16 cu. ft. per second normally against a flat vane. Find the pressure on the vane. If the vane moves in the same direction as the impinging jet, with a velocity of 32 ft. per second, find (a) the pressure on the vane; (ti) the useful work done ; (c] the efficiency. How would the results be affected if the vane were inclined at 45 to the jet, and moved in the direction of its normal with a velocity of 24 ft. per second ? Ans. 4000 Ibs.; 2250 Ibs., 72,000 ft.-lbs., ^; 1802.8 Ibs., 43,268 ft.-lbs., .169. 1 1. Two cubic feet of water are discharged per second under a pres- sure of 100 Ibs. per sq. in. through a thin-lipped orifice in the vertical side of a vessel, and strike against a vertical plate. Find the pressure on the plate and the reaction on the vessel. Ans. 475.82 Ibs. 12. A stream moving with a velocity of 16 ft. per second in the di- rection ABC strikes obliquely against a flat vane and drives it with a velocity of 8 ft. per second in the direction BD, the angle CBD being 30. Find (a) the angle between ABC and the normal to the plane for which the efficiency is a maximum; (b) the maximum efficiency ; (c) the velocity with which the water leaves the vane; (d) the useful work per cubic foot of water. Ans. (a] 21 44'; (d) .25664 ; (c) 12.6 ft. per sec.; (d} 256.64 ft.-lbs. 13. At 8 knots an hour the resistance of the Water-witch was 5500 Ibs.; the two orifices of her jet propeller were each 18 ins. by 24 ins. Find (a) the velocity of efflux ; (b) the delivery of the centrifugal pump; (c) the useful work done; {d} the efficiency; (e) the propelling H.P. assuming the efficiency of the pump and engine to be .4. Ans. (a) 29.4 ft. per sec.; (b} 1104.6 gallons per sec.; (c) 74,393 ft.-lbs.; (d) .63; (e) 536.7. 14. If feathering-paddles are substituted for the jet propeller in question 15, what would be the area of stream driven back for a slip of 25$? Find the efficiency and the water acted on in gallons per minute. Ans. 34.63 sq. ft.; .75 ; 234.206. 15. A jet issues horizontally, under a head of 20 ft., from a |-in. ori- fice in the vertical face of a tank and strikes normally the centre of a vane at a distance of 48 ins., measured horizontally, from the tank's face. By measurement the vertical distance of the point of impact, below the axis of the orifice, was found to be 2.582 inches. Find the coefficient of velocity (c v \ the inclination of the vane's axis to the horizontal, and the coefficient of impact (a) in the following cases : (a) A flat i2-in. circular vane, the balancing weight (W) being 3.015 Ibs. 410 EXAMPLES. (b) A hemispherical vane of 12 ins. diameter, W being 3.556 Ibs. (<:) A hemispherical vane of 3 ins. diameter, W being 5.776 Ibs. (d) A parabolic vane with a base of 12 ins. in diameter and 6 ins. in height, W 7 being 3.535 Ibs. (e) An elliptic vane, 6 ins. in height and having a base of 12 ins. diameter, W being 3.56 Ibs. The vane edge is inclined at 20 to the axis. Ans. .96411; 6 8'; (a) .9834; (b) -5799; (c) -94*9', (ft) .6086; (e) .5986. 16. Find the angle of blade at entrance, the useful effect, and the efficiency of a Borda turbine from the following data : the jet at entrance makes an angle of 60 with the horizontal ; the depth of the turbine is 3 ft.; the total fall to the point of discharge is 19 ft.; the mean diameter of the turbine is 4 ft.; the quantity of water passing through the tur- bine is 4cu. ft. per second ; the angle of blade at exit is 30. (Disregard hydraulic resistance.) Ans. 51 33'; 7.256 H.P.; .84. 17. What advantages are gained by increasing the depth and diam- eter of a Borda turbine and by curving the outlet lips of the buckets ? 18. A Borda turbine of 3.5 ft. mean diameter has a head of 12.96 ft. over the inlet, a practical efficiency of .75, a theoretic efficiency (i.e., dis- regarding hydraulic resistances) of .9265 and delivers 3 horse-power. The radial width of the water-passages is 3 ins., and the depth of the turbine is 2.04 ft. If there is to be no shock at entrance, find (a) the inlet and outlet lip angles ; (b) the velocity (v 9 ) of discharge ; (<:) the quantity of water used by the turbine. Ans. (a} ill 25', 25 12' ; (b} 8.4 ft. per sec.; (c) 2.532 cu. ft. per sec. 19. In a railway truck, full of water, an opening 2 ins. in diameter is made in one of the ends of the truck, 9 ft. below the surface of the water. Find the reaction (d) when the truck is standing; (b) when the truck is moving at the rate of 10 ft. per second in the same direction as the jet; (c) when the truck is moving at the rate of 10 ft. per second in a direction opposite to that of the jet. If this movement of the truck is produced by the reaction of the jet, find the efficiency. Ans. (a] 24.55 lbs - P er s q- '"? (^) 34-?8 Ibs. per sq. in.; (c) 14.3 Ibs. per sq. in.; .588. 20. From a ship moving forward at 6 miles an hour a jet of water is sent astern with a velocity relative to the ship of 30 ft. per second from a nozzle having an area of 16 sq. ins.; find the propelling force and the efficiency of the jet as a propeller without reference to the manner in which the supply of water may be obtained. Ans. 138^ ibs.; .4535. 21. A reaction wheel is inverted and worked as a pump. Find the speed of maximum efficiency and the maximum efficiency, the coeffi- cient of hydraulic resistance referred to the orifices being .125. Ans. Speed = twice that due to lift; .758. EXAMPLES. 22. A reaction wheel with orifices 2 ins. in diameter makes 80 revolu- tions per minute under a head of 5 ft. The distance between the centre of an orifice and the axis of rotation is 12 inches. Find the H.P. and the efficiency. Ans. .146 ; .596. 23. In a reaction wheel the speed of maximum efficiency is that due to the head. In what ratio must the resistance be diminished to work at f this speed, and what will then be the efficiency ? Obtain similar results when the speed is diminished to three fourths of its original amount. Ans. .94; .8896; 1.067; .75 24. In a reaction wheel, determine the per cent of available effect lost (i) if V* = 2gH\ (2) if V 1 = 4gH; (3) if F 2 = ZgH. What conclusion may be drawn from the results? Efficiencies are respectively .828, .9, .945. 25. A stream of 64 sq. ins. section strikes with a 4o-ft. velocity against -a fixed cone having an angle of convergence = 100; find the hydraulic pressure. Ans. 492.1 Ibs. 26. A jet of 9 sq. ins. sectional area, moving at the rate of 48 ft. per second, impinges upon the convex surface of a paraboloid in the direc- tion of the axis and drives it in the same direction at the rate of 16 ft. per second. Find the force in the direction of motion, the useful work done, and the efficiency. The base of the paraboloid is 2 ft. in diameter and its length is 8 ins. Ans. 25 Ibs.; 400 ft.-lbs. ; T |j. 27. A stream of water of 16 sq. ins. sectional area delivers 12 cu. ft. of water per second against a vane in the form of a surface of revolution, snd drives in the same direction, which is that of the axis of the vane. The water is turned through an angle of 60 from its original direction before it leaves the vane. Neglecting friction, find the speed of vane which will give a maximum effect. Also find impulse on vane, the work on vane, and the velocity with which the water leaves the vane. Ans. 36 ft. per sec. ; 562^ Ibs.; 20.250 ft.-lbs. ; 95.24 ft. per. sec. 28. A surface of revolution is driven in the direction of its axis with a velocity of 16 ft. per second by means of a jet of water of 18 sq. ins. sectional area, which moves in the direction of the axis with a velocity of 80 ft. per second, and impinges upon the convex side of the surface. The tangent at the edge of the surface makes an angle of 30 with the vertical. Find the pressure on the surface and the efficiency. Ans. 500 Ibs. ; .128. 29. A jet of water under a head of 20 ft., issuing from a vertical thin-lipped orifice i in. in diameter, impinges upon the centre of a vane } ft. from the orifice. Determine the position of the vane and the force of the impact (a) when the vane is a plane surface ; (b) when the vane is 6 ins. in diameter and in the form of a portion of a sphere of 6 ins. radius. Ans. (a) 13,679 Ibs. ; (&) 20,518 Ibs. or 6. 839 Ibs. according as vane is concave or convex. 412 EXAMPLES. 30. A stream of water i in. thick and 8 ins. wide, moving with a velocity of 18 ft. per second, strikes without shock a circular vane, of a length subtending an angle of 90 at the centre. The vane is driven in the direction of the stream with a velocity of 6 ft. per second. Find the pressure on the vane, the work done, and the efficiency. Ans. 22/-g Ibs. ; 93! ft.-lbs. ; J T . 31. A Pelton wheel of 2 ft. diameter makes 822 revolutions per min- ute under a pressure-head of 200 Ibs. per square inch, the delivery of water being 100 cu. ft. per minute. Fin.d the total H.P., assuming that the buckets are so formed that the water is returned parallel to its origi- nal direction, and leaves without energy. If the actual H.P. is 70.3, what is the efficiency ? Ans. 87.22 ; .805. 32. A vane moves in the direction ABC with a velocity of 10 ft. per second, and a jet of water impinges upon it at B in the direction BD with a velocity of 20 ft. per second ; the angle between BC and BD is 30. Determine the direction of the receiving-lip of the vane, so that there may be no shock. Ans. The angle between lip and BC 23 47'. 33. A jet moves in a direction ABC with a velocity v and impinges upon a vane which it drives in the direction BD with a velocity 4z/. The angle ABD is 165. Determine the direction of the lip of the vane at B, so that there may be no shock at entrance. Ans. The angle between lip and direction of stream = 14 3'. 34. The lip angle of a given bucket is 30, the relative velocity (} is one half the velocity (7/1) with which the water reaches the lip. If there is to be no " loss in shock," find the speed (u) of the bucket, the direc- tion (y] of the entering water, and show that if the speed is to be increased 10 per cent, the lip angle must also be increased by 55.6 per cent. Ans. y 15 31'. 35. A stream moving with a velocity v impinges without shock upon a curved vane and drives it in a direction inclined at an angle to the direction of the stream. The angle between the lip of the vane and the direction of the stream is x, and V is the relative velocity of the water with respect to the vane. If the speed of the vane is changed by a small amount, say n per cent, show that the corresponding change in the direction of the lip, in order that the water might still strike the n v . vane without shock, is -v^sm x. 100 V 36. A jet issues through a thin-lipped orifice i sq. in. in sectional area in the vertical side of a vessel under a pressure equivalent to a head of 900 ft. and impinges on a curved vane, driving it in the direc- tion of the axis of the jet. The water enters without shock and turns through an angle of 60 before it leaves the vane. Find (a) the speed of the vane which will give a maximum eflect ; (b) the pressure on the EXAMPLES. 4*3 vane ; (c) the work done ; (d) the absolute velocity with which the water leaves the vane; (e) the reaction on the vessel, disregarding contraction. Ans. (a} 80 ft. per sec. ; (6) 320.9 Ibs. ; (c) 46.68 H.P. ; (d) 184 ft. per sec. ; (e) 781.25 Ibs. 37. A stream of thickness / and moving with the velocity v im- pinges without shock upon the concave surface of a cylindrical vane of a length subtending an angle 2 at the centre. Determine the total pressure upon the vane (a) if it is fixed ; (b) if it is moving in the same direction as the stream with the velocity u. In case (d) also find (c) the work done on the vane. Ans. (a) 2--<*/v a sin a; (b) 2 bt(v uY sin a ; (c) 2-blu(v - u? sin 3 a. & S 38. A stream of water, 2 sq. ins. in sectional area, delivers I cu. ft. per second against the concave side of a hemispherical cup, which moves with a velocity of 20 ft. per second in the direction of the jet. Find the impulse, the work done, and the efficiency. 39. A curved vane subtends an angle of 90 at the point of intersec- tion of the normals at the two edges, and receives without shock a stream of water 2 ft. wide and \ in. thick, moving with a velocity of 20 ft. per second and driving the vane in the same direction. The actual direction of the water is turned through an angle of 45. Find (a) the speed of the vane ; (b} the velocity with which the water leaves the vane ; (c) the total pressure on the vane; (d) the efficiency. Ans. (a] 10 ft. per sec. ; (b} 14.14 ft. per sec.; (c) 23,017 Ibs. ; (d) .25. 40. A vane is in the form of the segment AB of a circle subtending an angle of 120 at the centre O. A stream of water, moving with a velocity v\ , strikes the vane tangentially at A and drives it in the same direction with a velocity n. Find the velocity (7/3) with which the water leaves the vane, and show that it leaves in the direction OB if 2/1 = \\u, and that the direction has turned through 90 if v\ = yi. Find the efficiency in the two cases, and show that v\ =.yt corresponds to maxi- mum efficiency. Ans. v) Breast Wheels, in which the water is received a little below the axis of rotation and acts partly by impulse and partly by its weight. (e] Overshot Wheels, in which the water is delivered nearly at the top and acts chiefly by its weight. 2. Undershot Wheels. Wheels of this class, with plane floats or buckets, are simple in construction, are easily kept in repair, and were in much greater use formerly than they are now. They are still found in remote districts where there is an abundance of water-power, and are also employed to work floating mills, for which purpose they are suspended in an open current by means of piles or suitably moored barges. They are made from 10 to 25 ft. in diameter, and the floats, which are from 24 to 28 ins. deep, are fixed either normally to the periphery of the wheel, or with a slight slope towards the supply-sluice, the angle between the float and radius being from 15 to 30. The depth of a float is from one fourth to one fifth of the radius and should not be less than from 12 to 14 ins. They are from 14 to 16 ins. apart, and generally from 416 UNDERSHOT WHEELS. 4*7 one half to one third of the total depth of float is acted upon by the water. Let Fig. 236 represent a wheel with plane floats working in an open current. FIG. 236. Let i\ be the velocity of the current. Let u be the velocity of the wheel's periphery. Let Q be the delivery of water in cubic feet per second. The water impinges upon a float, is reduced to relative rest, and is carried along with the velocity u. Thus and wQ the impulse (z/ t u) t ^ A f ~f>/4'~i ^ the corresponding useful work v being- ^ and the effi- D /^ , and the maxi- mum theoretic efficiency may be easily shown to be < .5. Ex. An undershot wheel with straight floats and weighing 15,000 Ibs. works in a rectangular channel with horizontal bed and of the same -width as the wheel, viz., 4 ft.; the stream delivers 28 cu. ft. of water per second, and the efficiency of the wheel is . Find the relation between the up-stream (v\) and down-stream () velocities. If the up-stream velocity is 20 ft. per second, find the down-stream velocity. If the diameters of the wheel and bearings are 20 ft. and 4 ins., respectively, and if the coefficient of friction is .008, determine the mechanical effect. 28 = 4-#i?/i or h\ and //, = -. z/i u Therefore the efficiency = (vi ) + ^(-^ . } = - * s>i* v 7'i 2 \?V V 3 2t( f 2/i + U\ I i(Vi )[ I 112 - r - r ] = -. Vi , \ vSu* I 3 If vi 20 ft. per second, then ;/ / 7 20 + \ i (20 )( I -- )=-. 200 \ 25 H / 3 It is found by trial that u lies between 5.9 and 6 and is very approxi- mately 5.97 ft. per second. The total available power = - . 28 . = 10937.5 ft. -Ibs. per. sec. Therefore the actual mechanical effect = -(10937.5) = 3645.83 ft.-lbs. per sec. The work absorbed by bearing friction = .008 x 15000 x 5.97 x = 11.94 ft.-lbs. per sec. The net delivery in ft.-lbs. = 3645.83 - 11.94 = 3633.89- LOSSES. 421 Losses. Four principal losses may be considered, viz. : (i) The loss of Q l cubic feet of the deeper fluid elements which do not impinge upon some of the foremost floats. According to Gerstner, t being the number of the floats immersed, and c being -J or $ according as the bottom of the race is straight or falls abruptly at the lowest point of the wheel. (2) The loss of Q 2 cubic feet of water which escapes between the wheel and the race-bottom. Approximately, the play at the bottom may be said to vary from a minimum, s l BC, when a float AB is in its lowest position, Fig. 238, to a maximum, B l C l CD > 2 C 2 , when c FIG. 238. tv two floats A^B^ , A Z B Z are equidistant from the lowest position,, Fig. 238. Thus the mean clearance + BD) = , nearly, being the wheel's radius. 422 LOSSES. 2 Tfy But - ! = distance between two consecutive floats n = 2 . B V D, very nearly, n being the total number of floats. Hence and therefore the mean clearance = s. -I ~. 4 ;/ 3 Again, the difference of head on the up-stream and down stream sides and the velocity of discharge, f rf , through the clearance is given by the equation Hence Introducing .7 as a coefficient of hydraulic resistance, I 7T 2 ' 4 n If the depth of the stream is the same on both sides of the wheel, i.e., if h v = 7/ 2 , then I'd = i' r (3) The loss of <2 3 cubic feet of water which escapes between the wheel and the race-sides. Let s 2 be the clearance on each side. Then <2 H = .7 X 2// t J 2 ?v = i-4^iVV ,7 being a coefficient of hydraulic resistance. MECHANICAL EFFECT. 42$ (4) Finally, if W\bs. is the weight on the wheel-journals, the loss due to journal friction yu being- the journal coefficient of friction, and p the journal radius. Actual Delivery. Thus the actual delivery of the wheel in foot-pounds Remarks. These wheels are most defective in principle, as they utilize only about one third of the total available energy. They may be made to work to somewhat better advantage by introducing the following modifications: (a] The supply may be so regulated by means of a sluice- board that the mean thickness of the impinging stream is about 6 or 8 ins. If the thickness is too small, the relative loss of water along the channel will be very great. If the thickness is too great, the floats, as they emerge, will have to raise a heavy weight of water. The sluice-board is inclined at an angle of 30 to 40 to the vertical, so that the sluice- opening may be as near the wheel as possible, thus diminishing the loss of head due to channel friction, and is rounded at the bottom to prevent a contraction of the issuing fluid. Neglect- ing frictional losses, etc., r i cc ^( TT , v \ * 2 \ ( Ioss f energy the useful effect = wQ\H -4- -- 1 1 , ig tgi \ due to shock l._^}_^Q^^f. 2g 2gl H being the difference of level between the point at which the water enters the wheel and the surface of the water in the tail- 424 MECHANICAL EFFECT. race, i.e., the fall. H is usually very small and may be nega- tive. If the vanes are inclined, the resistance to emergence is not so great, and the frictional bed resistance between the sluice and float is practically reduced to nil. With a straight bed and small slope (i in 10) the minimum convenient diameter of wheel is about 14 ft. (b) The bed of the channel for a distance at least equal to the interval between two consecutive vanes may be curved to the form of a circular arc concentric with the wheel, with the view of preventing the escape of the water until it has exerted its full effect upon the wheel. When the bed is curved, the minimum convenient diameter of wheel is about 10 ft. An undershot wheel with a curb is in reality a lo\v breast wheel, and its theory is the same. (<:) The down-stream channel may be deepened so that the velocity of the water as it flows away becomes > v r The impulse due to pressure is then positive, which increases the useful work and therefore also the efficiency. (d*) The down-stream channel may be widened and a slight counter-inclination given to the bed. What is known as a standing-wave is then produced, in virtue of which there is a sudden rise of surface-level on the down-stream side above that on the up-stream side. This allows of the wheel being lowered by an amount equal to the difference of level between the surfaces of the standing-wave and of the water-layer as it leaves the wheel, thus giving a corresponding gain of head. (e) The introduction of a sudden fall has been advocated in order to free the wheel from back-water, but it must be borne in mind that all such falls diminish the available head. 4. Poncelet Wheel (Figs. 239, 240). Thus undershot wheels with flat buckets have a small efficiency because of the loss of energy in shock at entrance and because of the loss of energy carried away 'by the water oh leaving the: wheel. These losses have been considerably modified in PONCE LET WHEEL. 425 Poncelet's wheel, which is often the best motor to adopt when the fall does not exceed 6J ft., and which, in its design, is FIG. 239. m// FIG. 240. governed by two principles that should govern every perfect water-motor, viz. : (i) That the loss of energy in shock at entrance shotild be- a minimum. 426 PONCE LET WHEEL. (2) That the velocity of the water as it leaves the wheel should be a minimum. The vanes are curved and are comprised between two crowns, at a slightly greater distance apart than the vane- width ; the inner ends of the vanes are radial, and the water acts in nearly the same manner as in an impulse turbine. A Poncelet wheel of from 10 to 13 ft. in diameter has 36 floats, while for wheels of from 20 to 23 ft. in diameter the number of floats is about 48. The wheels are usually from 10 to 20 ft. in diameter and have from 32 to 48 floats which may be of plate-iron or wood. First. Assume that the outer end of a vane is tangential to the wheel's periphery, that the impinging layer is infinitely thin, and that it strikes a float tangentially. Let of (Fig. 241) be a float, and aq the tangent at a. The velocity of the water relatively to the float = ?', u. The water, in virtue of this velocity, ascends on the bucket to a height / \2 pq = , then falls back and leaves the float with the relative velocity v^ u and with an absolute velocity v^ 2u. This absolute velocity is nil when the speed of the wheel is such that u = -J-z'j , and the theoretical height 2 of a float is pq = - 1 -. The total available head is thus 42- changed into useful work, and the efficiency is unity, or perfect. Taking R as the mean radius of the crown and u m as the corresponding linear velocity, the mean centrifugal force on u * each unit of fluid mass is -^- and acts very nearly in the direc- K tion of gravity, so that the height pq of a float may be approxi- mately expressed in the form I pq ~- PONCE LET WHEEL. 427 V being the velocity with which the water commences to rise on the float. Practically, however, the float is not tangential to the periphery at a, as the water could not then enter the wheel. Also, the impinging water is of sensible thickness, strikes the periphery at some appreciable angle, and in rising and falling on the floats loses energy in shocks, eddies, etc. Let the water impinge in the direction ac, Fig. 242, and take ac = v r Take ad in the direction of and equal to ?/, the .velocity of the wheel's pe- riphery. Complete the parallelo- gram bd. Then cd = ab = V is the velocity of the water rela- tively to the float. C| FIG. 242. FIG. 243. That there may be no shock at entrance, ab must be a tangent to the vane at a. Again, the water leaves the vane in the direction of ba produced, and with a relative velocity ac = ab = V. Complete the parallelogram de. Then ag (= v 2 ) is the absolute velocity of the water leaving the wheel. Evidently cdg is a straight line. Let the angle cad = y, and the angle bad = n a. From the triangle adc, V' 2 = 2v4 cos (0 428 PONCE LET WHEEL v* = F 2 + u* 2 Vu cos a ; . . . , (2) Sm v l sn a From the triangle v * = V* + u* + 2 Vu cos a ..... (4) By equations (i), (2), and (4), COS Therefore the useful work per second iv Q = 2u(v. cos y u} (5) g ivQ v 2 cos 2 Y This is a maximum and equal to - when v cos Y u = - - , and the maximum efficiency is cos 2 y. Hence fc too, the angle adb 90, and, by Fig. 243, bd 2pd tan (TT a) = - T = 2 tan y. . . (6) ad ad Also, V ab The efficiency is perfect if y is nil, and therefore a = 180. Practically this is an impossible- value, but the preceding cal- culations indicate that y should not be too large (usually < 30), and that the speed of the wheel should be a little less than one half of the velocity of the inflowing stream. Take y = 15 as a mean value. Then u fj X -484, and the efficiency = .993. PONCE LET WHEEL. 429 The best practice indicates the relation 112^= 2Ou. It must be borne in mind that the theory applies to one elemen- tary layer only, say the mean layer, and that all the other layers enter the wheel at angles differing from 15, thus giving rise to " losses of energy in shock." The losses of energy in frictional resistance, eddy motion, etc., in the vane-passages have also been disregarded. Tangential entrance is not possi- ble in practice and the efficiency does not exceed .65 for falls up to 4 ft., is .60 for falls of from 4 to 5.5 ft., and is from .55 to .50 for falls of from 5.5 to 6.5 ft. The greater efficiency of the Poncelet wheel, as compared with wheels having flat buckets, very clearly shows the importance of bringing the water on to the wheel in such a manner as to avoid loss of energy in shock and in the production of eddies. The layers of water, flowing to the wheel under an adjustable sluice and with a velocity very nearly equal to that due to the total head, may be all made to enter at angles approximately equal to 15, and the corresponding losses in shock reduced to a mini- mum by forming the course as follows: The first part of the course FG, Fig. 244, is curved in such FIG. 244. a manner that the normal pqr at any point / makes an angle of 15 with the radius oq. The water moves sensibly parallel to the bottom FG, and therefore in a direction at right angles 43 PONCE LET WHEEL. to pr. Hence at q the direction of motion makes an angle of 15 with the tangent to the wheel's periphery. If or is drawn perpendicular to pr, then or oq sin 15 = a constant. Thus the normal pqr touches at r a circle concentric with the wheel and of a certain constant diameter. The initial point F of the profile FG is the point in which the tangent to this circle, passing through the upper edge of the sluice-opening, cuts the bed of the supply-channel. Let d be the depth of the crown or shrouding, i.e., the normal distance between the outer and inner peripheries of the wheel. Let b be the width and / the thickness of the sheet of water entering the wheel. Then, disregarding the thickness of the floats, the capacity of the portion of the wheel passing in front of the entering stream per second is approximately bdu m . Practically, the whole of this space cannot be occupied by the water and mbdu m = Q = btv^ , m being a coefficient varying from \ to |-. Thus /, the thickness of the stream, = md v i ,R u = ma -- r i v i If the efficiency is a maximum, v l cos y = 2u, and then m R t = d cos y. 2 r l The head over the mean water layer at the point of entrance H being the available fall. Hence PONCELET WHEEL. 43 r an average value of c v being .9, and if, as according to Grashof, H= i6t, 32 Morin makes the radius (r^) of the wheel from two to three times the depth (d) of the crown, and Poncelet considers that IT TT this depth should be about and not less than . 3 4 In order, indeed, to prevent the water from rising over the top of the floats, d should be from to -//, and therefore r l from //to 2//, the latter being often adopted in practice. The area of the sluice-opening usually varies from 1.25^ to \.^bt. The inside width of the wheel is about (b -f- -J) ft. If \ is the angle subtended at the centre O of the wheel by the water-arc between the point of entrance a and the lowest point C, Fig. 245, of the wheel, and if aq' is drawn hori- zontally, then Aq' is approxi- mately the height of the float, and the theoretic depth ^ of the crown is given by d = AC Aq' + Cq' = Ag' + OC- Oq' V 2 = * 4- r,(i cos In practice it is usual to increase this depth by ^, the thick- ness of the impinging water-layer, and therefore 43 2 EFFICIENCY OF PONCE LET WHEEL. -cos The buckets are usually placed about I ft. apart, measured along the circumference, but the number of the buckets is not a matter of great importance. There are generally 36 buckets in wheels of 10 to 14 ft. diameter, and 48 buckets in wheels of 20 to 23 ft. diameter. It may be assumed that the water-arc is equally divided by the lowest point C of the wheel, so that the length of the water-arc = 2\r l = 2uT, T being the time of the ascent or descent of the water in the bucket. In the middle position, the upper end of the bucket should be vertical, and if the float is in the form of a circular arc, its radius ;' = d sec (n a), a being the angle between the bucket's lip and the wheel's periphery. The time of ascent or descent is also given by _ 9^ +_smjf> I i6 V where sin = -/cos (n a]. 2 x ' 5. Efficiency corresponding to a Minimum Velocity of Discharge (# 2 ). From Fig. 242, ao (= \ag) (V 2 ) sin Y sin aW ad Hence for any given values of u and y, v^ is a minimum when sin aod is greatest, that is, when aod = 90, or ag is at EXAMPLE. 433 right angles to de. Then also ad ae = ab, or u = F, and ac bisects the angle bad. Thus z\ = 2u cos y and v^ = 2u sin ^. The useful work W v = 2Z/ 2 COS 2V = - " " < The total available work Therefore the efficiency, 77, == - x - , cos 3 and the H.P. of the wheel = 550 Experience indicates that the most favorable value for u lies between .57^ and .6?^, and that the average value of the efficiency is about 60 per cent. Although, under normal conditions of working, the effi- ciency of a Poncelet wheel is a little less than that of the best turbines, the advantage is with the former when working with a reduced supply. Ex. To design a Poncelet wheel for a fall of 4* ft. and a water-supply of 24 cu. ft. per second, taking, as a first approximation, y = A = 20. Mean velocity (v\) at point of admission : vi = -9y ig - 4i = I 5- 3 2 9 ft. per sec. speed of periphery : Lip angle on : u = -i/i cos 20 = 7.06318 ft. per sec. tan (it a) = 2 tan 20 = .728, and it a. = 36 3', or a = 143 57'. Value of iff : -L sin - = Vcos 36 3' = .89917, and V> = i28".6' = 128. i. 434 EXAMPLE. Relative velocity ( V} at admission: V = u sec 36 3' = 8.7361 ft. per sec. Value of r\. Taking, as a first approximation, R = r\ = $d t Ui = u, and A = 20, then r' = d sec 36 3' = r\ x .4123, and 128.1 \. sin 128 6 20 y " ,80 / n x .4123 * 1 8o r ' ~7b~~ ~ y ~ (TT^Tsp ' which gives n 7.445 ft- or sa Y 7^ ft- Depth (d) of crown.- Taking, as an approximation, u\=u and R=r\ = T -7755 + / = 1.8 ft., suppose. More correct radius of float : r' = 1.8 sec 36 3' = 2.226 ft. Values of R and u\ : /e = 7. 5 --(i. 8) = 6.6 ft. , = u = 7. 06318 = 6,2156 ft. per sec. More correct value of A ; ^128.1 1 80 A . 7$ = 7.06318 sin 1 28 6' / 75 V " < 6 - ; V 32 + - '6 A/ 32 , < 6 - 2I 56) a or A = .298479, and A = 17.!. Thickness (/) ^?/" stream : 1 1.8 6.6 / = - COS 20 . = .372 ft. 2 2 7.5 Width (b) of wheel: b = = 4.29 ft. .372 x 15.0329 Time ( T] of ascent or descent of water on float : T=^ = .298479^^8 = -317 sees. Number of floats (N\ If spaced i ft. apart, N = 2Tt . 7! = 471, or, say, 48. FORM OF BUCKET. 435 Theoretical maximum power of wheel -^24(7.063 1 8) 2 = 4679.6 ft.-lbs. per sec. Total available power = 62^ . 24 . 4^ = 6750 ft.-lbs. per sec. Efficiency = = .693. 6. Form of Bucket. The form of the bucket is arbitrary, and may be assumed to be a circular arc. In practice there are various methods of tracing its form. METHOD I (Fig. 247). The tangent am to the bucket at a makes a given angle a with the tangent at a to the wheel's outer periphery. The radius of is also a tangent to the bucket at/. If the angle aofis known, the posi- tion of /on the inner periphery is at once fixed, and the form of the bucket can be easily traced. Let the angle aof = ,r. Join of and let the tangents to the bucket at a and f meet in m. Then the angle oam = a 90, " onia 1 80 - oam aom 270 a x, " mfa the angle maf = J(i8o fmd) FIG. 247. a X -45. Let r x , r 2 be the radii of the outer and inner peripheries of the wheel. Then r l oa sin ofa sin mfa r 2 ~ of~~ sin oaf ~~ sin oaf sin a x \ m( -45J since the angle oaf '== oam maf '= - 45 FORM OF BUCKET. Hence (a x\ la x\ -- 45 -f-| sin -- 45 -] _, _ f _ 2 ~^z) \2 ^ 2) r ' r * sin - 45 - 2 2 X tan - ( a x \ in(- 45 -- \2 / tan - ah equation giving x. The point o in which the perpendicular o'f to of meets the perpendicular o' a to am is the centre of the circular arc required, and o'f ( u - - = -4v, cos Y u}, 2/7" 2P" 2"' 7 ~ fr^ * i' 2 cos 2 y /* which is a maximum and = = , when 7' L cos y = 2u. In practice y is usually about 30, and 3? ; i 2 the maximum useful work = - Z2f' corresponding to the relation 4^= 1/37', , or // = .433^. To diminish as much as possible the loss in shock at V 2 entrance due to the dissipation of the energy in eddy motion, o the direction ab of the relative velocity V should be parallel to 444 MECHANICAL EFFECT OF BREAST WHEELS. the arm or tangential to the lip of the bucket and should there- fare be approximately at right angles to the wheel's periphery. If, at the point of entrance, the inlet lip is the lowest point of the bucket, the water flows upwards, and the relative velocity V, instead of being wholly destroyed in eddy motion, is par- tially destroyed by gravity. This latter is again restored to the water on its return, and increases the wheel's efficiency. For a given speed (#) of the wheel, the velocity (7^) with which the water should reach the wheel in order to make the y 2 loss of a minimum is found by making dV o in eq. (i), and then = 7^. dv l u cos Y - d v \ > or 7' t = u cos Y" This is an impossible relation, as it makes 7^ < ?/ and the useful work negative. In fact the angle afd (= baf ) in such case would be 90, and the direction af of v l would be prac- tically tangential, so that no water would enter the wheel. Again, for a given velocity 7^ of the water as it reaches the V* wheel, the speed of wheel which would make the loss of 2 2~ a minimum is given by o =. u . du 7^ cos Y du, or it = i\ c6s Y' This is also an impossible relation, as it makes the useful work nil. It will be found advantageous to use curved or polygonal buckets and not plane floats. A bucket, for ex- ample, may consist of three straight portions, ab, be, cd, Fig. 257. Of these the inner portion, cd, should be radial; the outer portion, ab, is nearly normal to the periphery of the wheel, and the central portion,, be, may make angles of about 135 with ab and cd. MECHANICAL EFFECT OF BREAST WHEELS. 445 Disregarding all other losses, the theoretical delivery of the wheel in foot-pounds = WQ where h 2 = total fall fall (h t ) required to produce the velocity v r If ?/ be the efficiency, then, according to the results of Morin's experiments, 77 = .40 to .45 if /^ = -H\ 4 V= .42 to .49if// 1= =|// ; ?t = .47 if ^ == 2 -H\ ?=.55 if //^l//. (V) There is a loss of head due to frictional resistance along the channel in which the wheel works. Let / = length of the channel (or curb). Let t = thickness of water-layer leaving the wheel. Let b breadth of wheel. The mean velocity of flow in this curb channel is approx- imately -u, and the loss of head due to channel friction f bt 2g ~ 3 bt 2' 9 where /= coefficiency of friction, b -\- 2t = wetted perimeter, bt = water area, and y being 30. (d) There is a loss of head due to the escape of water over the ends and sides of the buckets. Let s^ be the play between the ends of the buckets and the channel. 446 MECHANICAL EFFECT OF BREAST WHEELS. Let s. 2 be the play at the sides. (s l = s 2 , approximately.) Let s l , z. z , . . . z n be the depths of water in a bucket corresponding to n successive positions in its descent from the receiving to the lowest point. Let /j , /, . . . / be the corresponding water-arcs meas- ured along the wheel's periphery. The orifice of discharge at end of a bucket = bs r The mean amount of water escaping from a bucket over its end -f- . . c being the coefficient of discharge. The water escapes at the sides as over a series of weirs,, and the mean amount of water escaping from a bucket over the sides Hence the total loss of effect from escape of water per sec., h being the vertical distance between the point of entrance and the surface of the water in the tail-race (e) There is a loss of head due to journal friction. Let W= weight of wheel. Let w l = weight of water on the wheel. Let r l = radius of wheel's outer periphery. Let r' radius of axle. MECHANICAL EFFECT OF BREAST WHEELS. 447 Loss per second of mechanical effect due to journal friction r being the coefficient of journal friction. There is a loss of mechanial effect due to the resistance of the air to the motion of the floats (buckets), but this is prac- tically very small, and may be disregarded without sensible error A deepening of the tail-race produces a further loss of effect, and should only be adopted when back-water is feared. Hence the total actual mechanical effect, putting I U\ I V* U 2 -f 7.; * _ 2UV COS y\ is = wQ (H wQ[v - -+- .V 2J . n 2^ 2 I u(v, cos y-u) t W Q - bt 3 2g 5 n wZ\l - v?\ wQ - I + r^)+ u( Vl cos Y- +1L 1SL, - X W+ ,/). ^/ 3 2g ^ Hence for a given value of v v the mechanical effect (omitting the last term) is a maximum when v cos y u = - -- (= .433 X ^ , if y = 30). In practice the speed of the wheel is made about one half of the velocity with whicri- the water enters the wheel. For a given speed of wheel, and disregarding the loss of 443 EXAMPLE. u cos Y = O. effect due to curb friction, which is always small, the mechan- ical effect is a maximum for a value of i\ given by .~wZ\\ -f r . wQ n i g tl or u cos Y *! = The loss by escape of water, viz., c^2g , varies, on an average, from 10 to 15 per cent of the whole supply, so that c */2g varies from to . * n 10 20 Ex. The buckets of a low breast wheel, of 24 ft. diameter, are half filled with water which flows from a flume through a vertical rectangular sluice-opening at the rate of 15 cu. ft. per second. The linear speed of the wheel's periphery is 5 ft, per second. At the point of admission the inflowing jet has a velocity of 10 ft. per second and makes an angle of FIG. 258. 30 with the rim. The total available fall is 8| ft. Find (a) the position of the point of admission; (&} the work done by impact and weight; {*) the position and dimensions of the sluice-opening, the depth of the shrouding being 12 ins. S4GEBIEN WHEEL 449 (a) Let OB be the radius to the point of admission JB, and let be its inclination to the vertical. Draw the vertical OG and the horizontal BF. Theoretically, hi , the head required to develop a velocity of 10 ft. per second, ==".* Then 8| ~ i^ = 6|f ft. = head available for work by weight = the vertical fall on the wheel = FG. Therefore cos = -^ = -' ~ T * = .421875, and (p = 65 3', defining the position of B. (&) The useful theoretical work done by impact = -^'5 -5(io cos 30 - 5)= 536.i33ft.-lbs. The useful theoretical work done by weight = 62^ . 15 . 6\l = 6503.906 ft.-lbs., and the combined useful work = 7040.039 ft.-lbs. (c) Let AD, BD be the vertical and horizontal distances of the lift A from B. The angle between the direction of z/i at B and the horizontal == - 30 = 35 3' Therefore AD ^ sin 2 35 3' = .51533 ft. 04 and BD = 7 sin 70 6' = 1.4692 ft. Again, the width of the wheel = j = 6 ft., and the width of the sluice may be taken to be about 3 ins. less than this, or 5! ft. The head over the lip = I T 9 ^ 5 I 533 1.0472; the average velocity of flow through the sluice = . 9 4/64 x 1.0472 = 7.3656 ft. per second, and the depth of the sluice-opening = = -354 ft. II. Sagebien Wheels, Fig. 259, have plane floats inclined to the radius at from 40 to 45 in the direction of the wheel's rotation. The floats are near together and sink slowly into the fluid mass. The level of the water in the float-passages gradually varies, and the volume discharged in a given time may be very greatly changed. The efficiency of these wheels 45 OVERSHOT WHEEL. is over 80 per cent, and has reached even 90 per cent. The action is almost the same as if the water were transferred from FIG. 259. the upper to the lower race, without agitation, frictional resis- tance, etc., flowing away without obstruction into the tail-race. 12. Overshot Wheels. Since the introduction and develop- ment of the turbine these wheels have become almost obsolete. They have been considered among the best of hydraulic motors for falls of 8 to 70 ft. and for a delivery of 3 to 25 cu. ft. per second, and have proved especially useful for falls of 12 to 20 ft. The efficiency of overshot wheels of the best construc- tion is from .70 to .85. The thickness of the sheet of water passing through the sluice on to the wheel rarely exceeds 4 or 5 ins., and is often less than 2 ins. If the level of the head-water is liable to a greater variation than 2 ft., it is most advantageous to employ a pitch-back or high breast wheel, which receives the water on the same side as the channel of approach. 13. Wheel Velocity. This evidently depends upon the work to be done, and upon the velocity with which the water arrives on the wheel. Overshot wheels should have a low cir- OVERSHOT WHEEL. 45* cumferential speed, varying from 10 ft. per second for large wheels to 3 ft. per second for small wheels, and should not be less than 2|- ft. per second. At a higher speed than 6 ft. per second, if the buckets are more than two thirds full, the efficiency does not exceed 60 per cent. In order that the water may enter the buckets easily, its velocity should be greater than the peripheral velocity of the wheel. 14. Effect of Centrifugal Force. Consider a molecule of weight w in the * ' unknown ' ' surface of the water in a bucket (Fig. 260). At each moment there is a dynamical equilibrium between the "forces " acting on ;;/, viz. : (i) its weight w ; (2) the centrifugal the resultant T of the neighboring reactions. Take MF = w. MG = force -<&r\ (3) o and complete the parallelogram FG. Then J/// = T. The direction of T is, of course, normal to the surface of the water in the bucket. Let HM produced meet the ver- tical through the axis O of the wheel in E. Then MG G&r _ g FH OM MF " w and therefore MF OE' FIG. 260. taking g 32 ft. and n being the number of revolutions per minute. 45 2 OVERSHOT WHEEL. Thus the position of E is independent of r and of the posi- tion of the bucket, so that all the normals to the water-surface in a bucket meet in E, and the surface is the arc of a circle having its centre at E, or, rather, a cylindrical surface with axis through E parallel to the axis of rotation. 15. Weight of Water on Wheel and Arc of Discharge.- Let Q volume supplied per sec. , and N = number of buckets. Then - - = number of buckets fed per second, 271 and jrf- = volume of water received by each bucket per sec. NGO Hence the area occupied by the water until spilling com- mences = . , T , b being the bucket's width (= width of wheel between the shroudings). The water flows on to the wheel through a channel (Fig. 261), usually of the same width b as the wheel, and the supply is regulated by means of an adjustable sluice, which may be either vertical, inclined, or horizontal. When the water springs clear from the sluice, as in Fig. 261, the axis of the sluice should be tangential to the axis of the jet, and the inner edges of the sluice-opening should be rounded so as to eliminate contraction. Let j/, z be the horizontal and vertical distances between the sluice and the point of entrance. Let T be the time of flow between the sluice and entrance. Let v Q ,-v l be the velocities of flow on leaving the sluice and on entering the bucket. Then v l cos (y z^sin ( r and 6 being the angular deviation of the point of entrance from the OVERSHOT WHEEL. 45$ summit, and y the angle between the direction of motion of the water and the wheel at the point of entrance. If the bed of the channel is horizontal, and if also the sluice is vertical, opening upwards from the bed, and is of the same width b as the wheel, then FIG. 261. / being the depth of sluice-opening and h^ the effective head over the sluice. This effective head is about T 9 7 of the actual head. 32, -= = gives the delivery per foot Thus, taking g width of wheel. Taking .6 ft. and 3.6 ft. as the extreme limits between which /Zj should lie, and .2 ft. and .33 ft. as the extreme limits 454 OVERSHOT WHEEL. between which / should lie, then -7 must lie between the limits o 1.24 and 5, and an average value of ->- is 3. Thus the width of the wheel should be on the average _ . Again, disregarding the thickness of the buckets, the capacity of the portion of the wheel passing in front of the water-supply per second r a i r _ rf'Y ) I d\ - V = bd(*)\r^ -- 1 = bdr^c*), approximately, t r l being the radius and u the velocity of the outer circumference of the wheel, d the depth of the shrouding, or crown, and n the number of revolutions per minute. Only a portion, however, of the space can be occupied by the water, so that the capacity of a bucket is mubd, m being a fraction less than unity and usually or J. For very high wheels m may be -J-. Hence mbdu = Q. Therefore mdu =*-r. o The delivery (-7-) per foot of width must not exceed a certain limit, otherwise either d or u will be too great. In the former case the water would acquire too great a velocity on entering the buckets, which would lead to an excessive loss in eddy motion and a corresponding loss of efficiency ; while if the speed u of the wheel is too great, the efficiency is again diminished and might fall even below 40 per cent. The depth of a bucket or of the shrouding varies from 10 to 1 6 ins., being usually from 10 to 12 ins., and the buckets OVERSHOT WHEEL. 455 are spread along the outer circumference at intervals of 12 to 14 ins. The number of the buckets is approximately 5^ or 6r l , r l being in feet. The efficiency of the wheel necessarily increases with the number of the buckets, but the number is limited by certain considerations, viz. : (a) the bucket thickness must not take up too much of the wheel's periphery; (b) the number of the o ; FIG. 262. buckets must not be so great as to obstruct the free entrance of the water ;. (c) the form of the bucket essentially affects the number. Let the bucket, Fig. 262, consist of two portions, an inner 45 6 OVERSHOT WHEEL. portion bc y which is radial, and an outer portion, cd\ c being a point on what is called the division circle. The length be is usually one half or two thirds of the depth d of the shrouding. Take be = \d. It may also be assumed without much error that the water- surface ad is approximately perpendicular to the line cd, so that the angle eda is approximately a right angle. The spilling evidently commences when the cylindrical surface, having its axis at e and cutting off from the bucket a water-area equal to , passes through the outer edge d of the bucket. Let ft be the bucket angle cOd. Let be the inclination of Od to the horizon. Let be the inclination of ad to the horizon. Let r l be the radius of the outer periphery. Let R be the radius of the division circle. Let r 2 be the radius of the inner periphery. Then _g_ J0e _ cos (0 0) r^ Od ~ sin the sign being plus or minus according as the bucket is below or above the horizontal, and in the latter case, if 6 = 0, then /jCo 2 = g sin 0. Again, af = fd tan (0 -f- 0), approximately. Therefore the area dfa tan (6 + 0) = - tan (V + 0), where d = r l r y Hence the area abed = area cod area bof arpa dfa OYERSHOT WHEEL. 457 Equations (i) and (2) give 6 and 0, and therefore the posi- tion of the bucket when spilling commences. The bucket will be completely emptied when it has reached a position in which cd is perpendicular to a line from e to middle point of cd, or, approximately, when edc is a right angle. Let 6 l , 0j be the corresponding values of 6 and 0, and let y l be the angle between cd and the tangent at d to the wheel's periphery. Then and = 9 o - sn two equations giving L and # r Also, if ck is drawn perpendicular to dp > and N, the number of buckets, ^Fsin y l For efficient action the number of the buckets is much less than the limit given by this relation, often not exceeding one half of such limit. If y\ is very small, V = v^ u l , approximately, and there- fore md ^ T . sin v, /v. \ N<- ig-,). The capacity of a bucket depends upon its form ; and the bucket must be so designed that the water can enter freely and without shock, is retained to the lowest possible point, and is finally discharged without let v or hindrance. Hence flat buckets, Fig. 265, are not so efficient as the curved iron bucket in Fig. 268 and as the compound bucket made of three or two pieces in Figs. 266, 267, and 269. The resistance to entrance is least in the curved bucket, as there are no abrupt changes of direction due to angles. The capacity of a com- pound bucket may be increased, without diminishing the ease of entrance, by making the inner portion strike the inner periphery at an acute angle, Fig. 269. The objection to this construction, especially if the relative velocity V is large, is that the water tends to return in the opposite direction and escape from the bucket. FIG. 265. CAPACITY OF BUCKET. FIG. 266. 461 FIG. 267. / FIG. 268. FIG. 269. FIG. 270. 462 CAPACITY OF BUCKET. Let bed, efg, Fig. 270, represent two consecutive buckets of an overshot wheel turning- in the direction shown by the arrow. Water will cease to enter the bucket-space between bed and efg, and impact will therefore cease, when the upper parabolic boundary of the supply-stream intersects the edge d. The last fluid elements will then strike the water already in the bucket at a point M, whose vertical distance below d may be designated by z. The velocity i\' with which the entering particles reach M is given by the equation v {-_-_ \V*+2gZ ...... (I) Again, while the fluid particles move from d to J/let the buckets move into the positions d'c'b 1 ', e'f'g'. Let arc dd' = s^ = ee' . Let arc dM = s 2 . Let T be the time of movement from d to d' (or d to M}. Then s l = uT and assuming that the mean velocity from d to M is an arithmetic mean between the initial and final velocity of entrance. Thus Also, since the angle between */J/and the wheel's periphery is small, it may be assumed that the arc dM = de -\- ef-\- ee' , approximately, ^i ~ ^ ~" ^ y t v, u 27rr, v. u \ NOTE. ef ed ed^ - -^ . - , nearly. ] u u N u y ) EXAMPLES. 463 Thus and by equations (2) and (3), + 7'/ ~ 2* 2 r N u' an equation giving approximately the distance ^ passed through by a float during impact. The buckets can now be plotted in the positions they occupy at the end of the impact. The amount of water in each bucket being also known, the water-surface can be delineated, and hence the vertical distance z can be at once found. Ex. i. Find the angular depression of the water-surface below the horizontal (a) when the bucket lip is 37 14' above the centre, and (b) when the bucket lip is on a level with the centre; also find (c) the position of the bucket below the centre when a horizontal through the lip bisects the angle between the water-surface and the radius to the lip. The wheel has a diameter of 32 ft. and makes io revolutions per minute. The angular velocity ( = ~-r = . Then 7 60 10 -~, . . . sin 121 O) 32\io/ ~~ cos (37 14' 0) ~~ 200' , t 200 cos (37 14' 0) Therefore = = cos 37 H' cot $ + sin 37 14', 121 or cot = 1.316 and = 57 14'. 121 sin (0) = = tan = .60?, 200 cos (0+0) and = 31 10'. 121 sin sin sin to 200 cos (6 4- 0) cos 20 i2 sin a 0' IOO I or sin + sin = , 121 2 and sin = .4058, or = 23 56' = 6. 464 EXAMPLES. e Ex. 2. An overshot wheel has a diameter of 32 ft., a 12-in. crown, and its peripheral speed is 4 ft. per second. The lip of the bucket is j ins. thick. Water enters the wheel in a direction inclined at 60 to the vertical at a point 12 30' from the summit and with a velocity of 16 ft. per second. Spilling commences at 120 from the summit. Find (a) the relative velocity ( J 7 ) at admission ; (<$) the angle between the horizontal and the water-surface at i 47' 33'' above and at 30 below the cen- tre; (c) the angle (y<) between the bucket lip and the nm : (d} the point where the bucket is emp- tied ; (e) the bucket angle; (/) the elbow angle ; (g] the number of buckets ; (h) the bucket water area. At the point of admission d let dgh be the triangle of veloci- ties so that dg 1 6 ft., dh = 4 ft., and the angle gdh = 30 12 30' = 17 30'. Assuming also that the water enters without shock, the relative velocity V (dg) is parallel to the bucket arm cd t and the angle ^cdk ^3 angle ghk. Then FIG. 270. (a) F a = y^ 2 = dg* + dJi* ?. . gd . dh cos 17 30' = i6 2 + 4 2 2 . 16 . 4 cos 17 30' = 149.9242, and V = 12.2443 ft- P er sec - () When = i 47' 33", 32 = cos (i 47' 33" - 0) = i6( T \) 2 sin or cot =' 32 sec i 47' 33" - tan i 47' 33" and 0=i 47' 33". When B = 30, cos (30 + 0) sin = 32, EXAMPLES. or cot = 32 sec 30 + tan 30 = 37.5 2 77 and = i 31' 24". sin 17 30' _ 12.2443 sin yi 10 12.2443 or cosec yi ^~ cosec 17 30 ' = 2.545, and y l = 23 8'. (d) At the point // s , where the spilling is completed, Od z k 9 is a right angle and the angle Od*e = angle c*d*k* = y\ Then sin 0i ed* r\ i6/4\ 2 i sin yi ~ oe ~~ g ~~ 32 \i6/ ~ 32' sin 23 8' or sin 0i = = .0122772, 3 2 and 0i = o 42'. Therefore GI = 90 (7, + 0,) = 66 10', and the bucket is emptied at 90 + 66 10' = 156 10' from the summit, (f) n = 1 6'; A 3 = 1 5V. Therefore tan 23 8' = tan yi = - = .4272, This last equation is easily reduced to the form cos 2 ft 1.7458 cos fj = .74674, and cos ft .9962, or ft = 5 8', 121 . = - in circular measure. 1350 (/) The elbow angle Ocd 180 ft Ode = 18 - ft (90 ^i> = 90 - 5 8' + 23 8" = 108. 44 or N = 79.8, say 80. An empirical approximate rule makes 44 16 466 EXAMPLES. (h) - sin 5 8' - = tan 30 + water area of bucket* Therefore the water area = 11.09475 10.08333 ~" .28867 = .72275 sq. ft. = 104 sq. ins. Ex. 3. An overshot water-wheel, of 40 ft. diameter, is 12 in. wide and lias a 9.6-in. shrouding. The pitch-angle is 4 and the thickness of the bucket lip is i in. At the point where spilling commences the bucket water area is 24-^ sq. ins. Find the number of buckets, the point where spilling commences, and the angle between the rim and the bucket lip. r, = 20 ft.; 7? = 20 - - ; = 19.6 ft.; r-i 20 .8 = 19.2 ft. Take /3, the bucket-angle, = --4 = 5. s I .8 TO. 2 Then 19 . 2 x 7f~ + tan = 27t-^, i 80 12 2 A 20 X 19.6 . (IQ.2) 2 5 (.8) 3 24^ and - sin 5 - ie-%- = ------ tan 6 + -. 2 2 1 80 2 144 Hence N 164.5, and tan 9 2.56, or = 68 40'. The empirical formula gives 2itfi 44 20 Again, tan yi = - > cosec 5 cot 5 = .27782, and yi = i$ 32'. Ex. 4. One foitrtJi of the theoretic capacity of a bucket is filled with water. The angle between the bucket lip and the wheel's periphery is 20, the radius to the outer periphery is 18 ft. and the depth of the crown is 12 ins. If the velocity of the water at entrance is twice that of the wheel's periphery, find the greatest number of buckets theoretically possible. The number < 103.2. The actual number may be about two thirds of this, or 69. MECHANICAL EFFECT OF OVERSHOT WHEELS. 467 16. Useful Effect. (a) Effect of Weight. 1\\z wheel should hang freely, or just clear the tail-water surface, and the total fall is measured from the surface of the water in the tail- race to the water-surface just in front of the sluices through which the water is brought on to the wheel. Let h l , Fig. 272, be the vertical distance between the FIG. 272. centres of gravity of the water-areas of the first and last buckets before spilling commences. Then hj R cos 6 + r l sin 6, very nearly. Let // 2 be the vertical distance between the centres of gravity of the water-area of the bucket which first begins to -spill and the point at which the spilling is completed. Then h 2 = r^sin t sin 0), very nearly. The useful work per sec. = wQ(h t -j- kh 2 ), k being a frac- tion < I and approximately = .5. Let A Q be the water-area in the bucket which first begins to spill. 468 MECHANICAL EFFECT OF OVERSHOT WHEELS. Between this bucket and the one which is first emptied,, i.e., in the vertical distance k 2 , insert s buckets, at equal dis- tances apart, and let their water-areas A I} A 2 , A 3 , . . . A s be carefully calculated. Let Q m be the mean amount of water per bucket in the discharging arc. Let A m be the mean water-area per bucket in the discharg- ing arc. Then A 9 + A l + A,+ .. .+A,_ t + A s A <- The value of can now be easily .found, since k - - - ^ ' Q " A t - Let q be the varying amount of water in a bucket from which spilling is taking place, and at any moment let y be the vertical distance between the outer edge of the bucket and the surface of the water in the tail-race. q is a function of y and depends upon the contour of the water in the bucket- Let Y be the mean value of ^ y between the points where spilling begins and ends, i.e., for values r { and y., of y. Then since r / q . dy. r r I y . dq = yq - - I q ./ is Again, the elementary quantity of water, dq, having an initial velocity equal to that of the wheel, viz., u, falls a dis- tance y and acquires a velocity = Vu z -f- 2gy. MECHANICAL EFFECT OF OVERSHOT WHEELS. .469 Thus it flows away in the tail-race, causing a loss of w . dq energy = ~y-(^ 2 + 2 T) = - Hence the total loss of energy between the points where spilling begins and ends Overshot and pitch-back wheels do not work well in back- water, as they lift a greater or less weight of water in rising above the surface. If the water-level in the race is liable to variation it is better to diminish the diameter of the wheel and design it so that it may never be immersed to a greater depth than 12 ins. (b) Effect of Impact. The head //' required to produce the velocity v^ with which the water reaches the wheel is theoreti- cally l ; but as there is a loss of at least 5 per cent in the 7/2 most perfect delivery, it is usual to take //' = v , an average value of v being I . I . Let the water enter the bucket in the direction ac, Fig. 273. Take ac = i\. The water now moves round with a velocity u (assumed the same as that of the division circle), and leaves the wheel with the same velocity. Take ab in the direction of the tangent to the division circle at the point of entrance = u. The component be represents the relative velocity V of the water with respect to the bucket, and this velocity is wholly destroyed, ab must necessarily be parallel to the outer arm of the' bucket, so that there may be no loss, of shock at entrance. Then the impulsive effect ' __ - ' 2 But MECHANICAL EFFECT OF OYERSHOT WHEELS. ^u cos y -y being the angle through which the water is deviated from its original direction at the point of entrance. . 273. Hence the impulsive effect wQ = ruto cos y - u), o and the TOTAL USEFUL EFFECT wO ^u(v 1 cos y u) loss due to journal friction. o Designating the first two terms of this expression by P, the loss due to journal friction = JLll+ W\-U, ( U ) ^l p being the radius of the axle, and ^Fthe weight of the wheel. EXAMPLE. 47* Ex. An overshot wheel weighing 20,000 Ibs., with a 12-in. crown and of 40 ft. diameter, receives 400 cu. ft. of water per minute and revolves in 6-in. bearings. The water enters the buckets at 12 from the wheel's summit, with a velocity of 16 ft. per second and at an angle of 10 with the wheel's periphery, which moves with a linear velocity of 9 ft. pet- second. Spilling commences and is completed at points which are respectively 140 and 160 from the wheel's summit. Determine the power of the wheel and its efficiency, taking /- = .5 and ja =.04. Take A' = radius of division circle = 19^ ft. Then h\ = 19^ cos 12 + 20 cos 40 34.3947662 ft., and hi = 20 cos 20 20 cos 40 = 3.472964 ft. Therefore the H.P. due to weight 62^ . 400/ i f \ 33000 V 34 ' 3947662 + - x 3-472964J = 27.37215, and the H.P. due to impact 62$ 400 = . - 9(16 cos 10 9) 32 33000 ^ = 1.43968. Again, the weight of the water on the wheel i 20 \ - . - 2 i so) = 2231.04 Ibs., approx., 400. 62^/ 128 = - 207T + 207T . - . - 60.9 v i so and the total weight on the axle = 22231.04 Ibs. Thus the energy absorbed by frictional resistance in H.P. 22231.04 i = - x .04 x 9 = .18189, 550 V and hence the net useful work in H.P. = 27.37215 + 1.43968 .18189 = 28.62994. The total available H.P. . 400 / i6 2 33000 - + 20 COS 12 + 20 = 33.0022, )- 28 6^994 and therefore the efficiency = 222. .8675. 47 2 PITCH-BACK WHEEL. 17. A pitch-back or high breast wheel is to be preferred to an overshot wheel when the surface-levels of the head- and tail-water are liable to very considerable variation. In the pitch-back wheel the water is admitted by aft adjustable sluice into the buckets on the same side as the supply-channel, Figs. 274 and 275. Thus the wheel revolves FIG. 274. FIG. 275. FIG. 276. in the direction in which the water leaves, and the drowning of the wheel is prevented. Further, the buckets may be now ventilated, Fig. 277, and may therefore be placed closer together than in the un ventilated overshot wheel. The efficiency of the pitch-back is at least equal to that of the overshot. EXAMPLES. 473 EXAMPLES. 1. An undershot wheel works in a rectangular channel 4 ft. wide, in which the water on the up-stream side is 2 ft. deep and flows with a velocity of 12 ft. per second ; the water on the down-stream side is 3 ft. deep. Find the useful work done and the efficiency. Ans. 1000 ft.-lbs.; / T . 2. Determine the maximum mechanical effect of an undershot wheel of 12 ft. diameter making 10 revolutions per minute, the fall being 3 ft. and the quantity of water passed per second 15 cu. ft. Ans. 1423 ft.-lbs. 3. Ascertain the general proportions of a Poncelet wheel, being given : height of fall = 4^ ft.; delivery of water = 40 cu. ft. per second ; radius of exterior circumference = 9 ft.; y 20. Ans. a. = 143 57'; ip = 128. i ; d =. 2. ft.; ;-' = 2.47 ft.; A. = 1 5.2 ; / = 5 ins.; N = 57 ; tj .69. 4. Design a Poncelet wheel for a fall of 4.5 ft. and 24 cu. ft. of water per second, using the formulae on pages 428-432, taking y = 20, and also A i= 20 as a first approximation. Ans. <* = 143 57'; depth of crown = 1.8 ft.; depth of stream = .372 ft ; = 4.14 ft.; radius of bucket 2.26 ft.; ^ = 128 6'; A = 17 i'; number of buckets = 48 ; mechanical effect = 8.5 H.P.; efficiency = .69. 5. An undershot water-wheel with straight floats weighing 1 5,000 Ibs. works in a straight rectangular channel of the same width as the wheel, viz., 4 ft.; the stream delivers 28 cu. ft. of water per second, and the efficiency is ^. Find the relation between the up-stream and down- stream velocities. If the velocity of the inflowing water is 20 ft. per second, find the velocity on the down-stream side and determine the mechanical effect of the wheel, its diameter being 20 ft., the diameter of the gudgeons being 4 ins., and the coefficient of friction .008. Ans. 3634.06 ft.-lbs. 6. Determine the effect of a low breast or undershot wheel 15 ft. in diameter and making 8 revolutions per minute, the fall is 4 ft. and the delivery 20 cu. ft. per second ; the velocity of the stream before coming, on the wheel is double that of the wheel. Ans. 3148 ft.-lbs. 7. 20 cu. ft. of water per second enter an undershot wheel of 30 ft. diameter, making 8 revolutions per minute, through an underflow sluice. The velocity of the entering water is twice that of the wheel's periphery. Find (d) the head of water behind the sluice; (b} the fall; (c) the theoretical mechanical effect; (d) the actual mechanical effect, disre- garding axle-friction. 474 EXAMPLES. Ans. (a) 2.716 ft.; (&) 1.283 ft -; fc) 5-7 2 H.P.; (= 5 51'; f But u x = roo\ tan 6 r-^\ tan 0' = -r~- , since M'N is necessarily tangential to the actual path RM' at M' \ and Aj.Vj.cos 6 = Q, the volume of flow per second, A x being the sectional area of the passage at right angles to OM. Substi- tuting these values in the last equation, GO d ^ A ^--^r^ and therefore TJ being the internal radius of the wheel. But the expression /* Adr is the volume of the passage t/r, between aa' and M and may be determined by actual measure- ment. Designating this volume by U x , then, an equation giving when r is known, or 0' when is known. Thus the actual position of the particle can be determined if its relative position is known, or its relative position can be found when its actual position is given. 490 CLASSIFICATION OF TURBINES. Take a number of equidistant points M { , M 2 , M 3 . . . along the axis of the passage, and let X , 2 , 3 . . . be the angular deviations of OM l , OM 2 , <9J/ 3 . . . from OR. Also, let /! , /2 , 7 3 . . . be the volumes of the passage between ##' and M l , ##' and J^, ##' and J^ . . . Then the angular deviations 0/, 2 ', 3 ' ... of the radii to the corre- sponding points J/j', J/ 2 ', J/ 3 ' ... on the actual path, are given by the equations and the actual path can be at once plotted. The value of / A x dr can easily be found graphically. tA-j Thus, plot the radii OR, OM lt OM 2 . . . OM as abscissae, and the corresponding sectional areas of the passage at -R, M l , .M 2 ... J/ as ordinates. Joining the upper ends of these ordinates by a suitable curve, the area between this curve, the extreme ordinates and the line of abscissae is evidently the volume required. This area may be determined with a pla- nimeter. 3. Classification of 'Turbines. The character of the con- struction of turbines has led to their being classified as (i) Radial-flow turbines; (2) Axial-flow turbines; (3) Mixed- flow turbines. The water may act wholly by pressure or who 11}' by im- pulse, or partly by pressure and partly by impulse, or by reaction. In pressure wheels the water-passages are not com- THE FOURNEYRON TURBINE. 49* pleteiy filled as in reaction wheels. In impulse wheels the Water spreads out in all directions, while in pressure and reaction wheels the water flows off on one side only. In Radial- flow turbines the water flows through the wheel in a direction at right angles to the axis of rotation and approximately radial. The two special types of this class are the Outward- flow turbine, invented by Fourneyron, and the Inward-flow or Vortex turbine, invented by James Thomson. In the outward-flow turbine. Figs. 285 and 286, the water enters a cylindrical chamber and is led by means of fixed guide-blades outwards from the axis. It is distributed over the inlet-surface, passes through the curved passages of an annular wheel closely surrounding the chamber, and is finally discharged at the outer surface. The wheel works best when it is placed clear above the tail-water. A serious practical defect is the difficulty of constructing a suitable sluice for regu- lating the supply over the inlet-surface. When the water is insufficient to work the turbine at its full power, the exit openings may be closed to any required extent by lowering a cylindrical sluice. A well-designed turbine of this type gives an efficiency of 70 per cent, and the maximum efficiency is about 80 per cent, but the efficiency is considerably diminished by closing the sluice. Fourneyron was led to the design of this turbine by observing the excessive loss of energy in the ordinary Scotch turbine, or reaction wheel, and introduced guide-blades in order to give the water an initial forward velocity and thus cause a diminution of the velocity of the water leaving the outlet-surface. Boyden's turbine is a modification of the Fourneyron. The water is conducted to the guide-blades, which are inclined so as to receive the water tangentially, through a truncated cone; and the water thus acquires a gradually increasing velocity together with a spiral motion. The wheel, again, is surrounded by a diffusor which expands outwardly and which 492 THE BOY DEN TURBINE. should be completely submerged. The water then flows through the wheel with an increased velocity and passes away FIG. 285. FIG. 286. through the diffusor with a velocity which gradually diminishes. There is said to be a gain of 3 per cent effected by this arrange THE YORTEX TURBINE. 493 FIG. 287. ment, while Boyden claimed for his 75 -H. P. turbine an effi- ciency of 88 per cent. In the Inward-flow or Vortex turbine, Figs. 287, 288, and 289, the wheel is en- closed in an annular space, into which the water flows through one or more pipes, and is usually distributed over the inlet-surface of the wheel by means of four guide-blades. The water enters the wheel, flows towards the space around the axis, and is there discharged. This tur- bine possesses the great advantage that there is ample space outside the Thomson's Vortex Turbine. '///^///////"""W FIG. 288. FIG. 289. wheel for a perfect system of regulating-sluices. This turbine has attained an efficiency of 77^ per cent. 494 THE AXIAL-FLOW TURBINE. Axial-flow turbines, Fig. 290, are also known as Parallel and D(nvnward-flow turbines and are sometimes called by the names of the inventors, Jonval and Fontaine. In these the water passes downward through an annular casing in a direc- tion parallel to the axis of rotation, and is distributed by means of guide-blades over the inlet-surface of an adjacent wheel. It enters the wheel-passages and is finally discharged vertically, or nearly so, at the outlet-surface. The sluice-regulations are worse even than in the case of an outward-flow turbine, but there is this advantage, that the turbine may be placed either below the tail-water, or, if supplied with a suction-pipe, at any point not exceeding 30 ft. above the tail-water. FIG. 290. If a turbine is designed so that the pressure at the clearance between the casing and the wheel is nil, and with curved passages in the form of a freely deviated stream, it becomes what is called a Limit turbine. In its normal condition of working it is an Impulse turbine, but when drowned it is a Reaction turbine, with a small pressure at the clearance. For moderate falls with a varying supply its average efficiency is higher than that of a pressure turbine. The Mixed- or Combined-flow (Schiele) turbine is a com- bination of the radial and axial types. The water enters in a THE AXIAL-FLOW TURBINE. 495 nearly radial direction and leaves in a direction approximately parallel to the axis of rotation This type of turbine admits of a good mode of regulation and is cheap to construct. The Swain turbine is a combination of the inward- and axial-flow types. The vane inlet-lips are vertical opposite the guide-blades, and at the outlet the vanes are bent into a quadrant of a circle. An efficiency of 88 per cent has been claimed for this turbine under a full load. Comparison of Outward- flow Turbines. Fourneyron deals with a varying supply of water by means of a circular sluice, which can be made to close off any required portion of the wheel. A similar arrangement may be added to the Cadiat turbine, which is of the outward-flow type and is fed from above through a cylindrical reservoir, the upper and lower edges of the reservoir being rounded to diminish the loss due to contraction. The objection to sluices of this kind is that the passages no longer run full when the inlet orifices are partially closed and there is therefore a considerable diminution of efficiency. In the Whitelaw turbine, p. 375, this difficulty can be obviated by changing the outlet instead of the inlet area. The absence of guides in the Cadiat and Whitelaw turbines make their construction somewhat simpler, but their efficiency is comparatively small, that of the Cadiat being about 65 per cent, while the efficiency of the Whitelaw turbine varies from 50 to 60 per cent. On the other hand, the Fourneyron turbine has an efficiency of more than 70 per cent and is mechanically a much more perfect machine. The guides in the turbine render it possible to utilize almost the whole of the energy of the water either by equalizing the peripheral and relative speeds at the outlet, or by making the absolute velocity at the outlet radial. The Fourneyron and Cadiat turbines are specially adapted for a large supply of water and a moderate fall, say not exceeding about 30 ft., while the Whitelaw tur- bines are found more useful for a small supply of water and a high fall. 49$ THEORY OF TURBINES. FIG. 291. Enlarged Portion of Section through XV, Fig. 287. FlG. 292. Enlarged Portion of Section through XY, Fig. 285, -M, m FlG. 293. Enlarged Portion of a Cylindrical Section JTr, Fig. 290, Developed in Plane of Paper. THEORY OF TURBINES. 497 4. Theory of Turbines (Figs. 291, 292, and 293). Denote inward-flow, outward-flow, and axial-flow turbines by I. F., O. F., and A. F., respectively. Let j\ , r 2 be the radii of the wheel inlet- and outlet-surfaces of an I. F. or O. F, Let r^ , r 2 be the outer and inner radii of the wheel inlet- surface of an A. F. / r _[__ r \ Let R be the mean radius (= -) of an A. F., assumed constant throughout. Let A l , A 2 be the areas of the wheel inlet- and outlet- orifices. Let d l , d 2 be the depths of the same in an I. F. or O. F. Let d v , d 2 be the widths of the same in an A. F. Let // be the depth of the wheel in an A. F. Let H l be the effective head over the inlet-surface of the wheel. This is the total head over the inlet- surface diminished by the head consumed in frictional resistance in the supply-channel, and by the head lost in bends, sudden changes of section, etc. Then H \ + h is the total head over the outlet of an A. F. available for work. Let H 2 be the fall from the outlet-surface to the surface of the water in the tail-race. If the turbine is submerged, then 7/ 2 is negative. Let i\ , ?' 2 be the absolute velocities of the water at the inlet- and outlet-surfaces. Let z/j, u 2 be the absolute velocities of the inlet- and outlet- surfaces. In an A. F. turbine i^ u 2 . Let V l , V 2 be the velocities of the water relatively to the wheel at the inlet- and outlet-surfaces. Let the angular velocity of the wheel = &? = = . r \ r z Let rf designate the hydraulic efficiency of the turbine. 498 THEORY OF TURBINES. Let the water enter the wheel in the direction ac, making an angle y with the tangent ad. Take ac to represent v^ , and ad to represent u r Complete the parallelogram bd. The side ab represents V l , and in order that there may be no shock at entrance, ab must be tangential to the vane at a. Again, at f draw fg, a tangent to the vane, and//, a tangent to the wheel's periphery. Takey^- and// to represent F 2 and u 2 respectively. Com- plete the parallelogram gk. The diagonal fh must represent in direction and magnitude the absolute velocity v 2 with which the water leaves the wheel. Let the angle hfk = #. Draw cm perpendicular to ad* and hn perpendicular to gk. The tangential component, viz., am or fn, of the velocity of the water as it enters or leaves the wheel is termed velocity of whirl (v^). The radial component, viz., cm or hn, of the velocity of the water as it enters or leaves the wheel is termed velocity of flow (y r \ Take vj = am, v w " fn, v r f cm, v r " h n * Let the angle bad 180 a. Let the angle gfk 180 ft. Thus a and ft are the angles which the vane (or blade) tips (or lips) make with the wheel's peripheries. Then, at the inlet-surface, v w ' j= i\ cos y accos y = am = ad dm = u l V l cos a, (i ) v r ' = i\ sin y = cm = V l sin a; . (2) .and at the outlet-surface, v w " = v 2 cos d fn fk kn u 2 V z cos ft, . (3) v r ff = v 2 sin d. = hn = F 2 sin ft . (4) THEORY OF TURBINES. 499 Let Q be the quantity of water which passes per second through the turbine. Then, disregarding the thickness of the vanes, in an I. F. or O. F. turbine V^ = V r ' - 271T& = Q = V#" . 27rr 2 d 2 = V r "A 2 , . (5) and therefore ' ' Also, if d^ d 2 d, ' In an A. F. turbine v r / A 1 = v r ' . 27rR . d, = = v r " . 2^R . d 2 = v r /7 A 2 , (6) and therefore " Q = v Allowance may be made for vane thickness as follows : Let 6 be the angle between the vane of thickness BC and the wheel's periphery A B. Then the space occu- pied by the vane along the wheel's periphery is AB= BC cosectf. Let n be the number of the guide-vanes, and t their thickness. FIG. 294. Let n l be the number of the wheel-vanes, and t ir t 2 their thickness at the inlet- and outlet-surfaces respec- tively. Then, in a radial-flow turbine, A l = ^{2777^ nt cosec y n l t l cosec a} . (7) THEORY OF TURBINES. and - cosec T 9 being a fraction depending on practical considerations. In an axial-flow turbine R is to be substituted for r l and r 2 in the values of A l and A 2 . n^ may be made equal to n + I or n -f- 2. Work and Efficiency. As the water flows through the wheel, let v be the velocity of flow at any point N distant r FIG. 295. (= ON} from the axis O^ and let / be the length of the per- pendicular from O upon the direction of v. Then v momentum of moving mass of water g = impulse on wheel = Fj suppose. Therefore, also, vp =. Fp moment of couple producing rotation, e> and the useful work of the couple per second wQ =. rpoo = vpQD. THEORY OF TURBINES. 501 But if v m is the component of v at N perpendicular to the radial line ON, vp v w = v cos B = , and therefore the useful work of the couple per second wQ W. Thus in an I. F. or O. F. turbine the useful effect at inlet = -^-voo = -^-v'u the useful effect at outlet = o o <5 and the USEFUL WORK per second done by the water on thw wheel between inlet and outlet wQ y (Or, -~ v w 'r 2 )c*, ..... ( 9 > wO The EFFICIENCY is given by the relation rj X wQH^ = the useful work per sec. wQ ~(^w\ - VX)> or ijg^ = V M , / U I - v^X , ..... (ii] which is the fundamental equation governing the design of I. F. or O. F. turbines. In an A, F. turbine wQ wQ the useful effect at inlet = - vJRoo vju, o o the useful effect at outlet = ^v^'Roo vju lt o o 5 2 THEORY OF TURBINES. and the USEFUL WORK per second done by the water on the wheel between inlet and outlet w ' - Y W ")R<, ........ (12) = "JW - T."X ....... ('3) The efficiency is given by the relation rj X wQ(H l -f- //) = the useful work per sec. or ^g(H 1 + h)-(v / -v w ")u 1 , .... (14) which is the fundamental equation governing the design of A. F. turbines. Again, disregarding hydraulic resistances, each pound of v * water on leaving the turbine carries away ft.-lbs. of energy. Hence the USEFUL WORK in an I. F. or O. F. turbine V * the corresponding EFFICIENCY being I -- ~, . . (16) and the USEFUL WORK in an A. F. turbine 1 + h- ..... (17) the corresponding EFFICIENCY being I - _ . 2 ... (18) o \ 1 I / THEORY OF TURBINES. Assuming that the velocity of whirl at outlet, viz. , v w " ', is nil and that H is the portion of ff l , or of H l -f- h, which is transformed into useful work, then gH u^J = ufa v r ' cot a), which may be written in the form I = a quadratic giving t V cot or // z/,/ \ 2 cot 2 a 7w ~~ ' ^Tff ~^ V ^ ~^W> ~T~ This result has been employed in preparing the following Table of values of - 1 corresponding to different values of ^= and of a : VgH *" 15 30 45 60 75 90 105 120 135 150 165 1.0 3.983 2.189 1.618 329 .142 I .874 752 .618 457 .251 0.9 3.629 2.047 1-547 .188 .128 I .887 773 647 .488 .271 0.8 3.289 .909 477 .090 .114 .900 795 .677 524 304 0.7 2.952 .776 .409 .225 095 .908 .819 .709 .563 339 0.6 2.621 .647 344 .188 .083 .923 .842 744 .607 .382 0.5 2.301 .523 .281 155 .069 935 .886 .781 657 -435 0-45 2.145 463 .250 139 .062 .942 .879 .800 .683 .465 0.4 .991 .405 .220 .122 055 949 .890 .820 .712 499 0-35 .847 346 .190 .107 .048 954 .902 .840 -744 541 o-3 .705 293 .161 .090 .041 .961 .917 .860 773 .586 0.25 .569 .240 .132 .074 034 .967 930 .882 .806 .636 0.2 .440 .188 .105 .059 .027 973 943 90S .842 .694 0.15 .318 .138 .078 .044 .O2O .980 .966 .927 .878 758 O. IO .204 .090 .051 .029 .013 .987 .972 951 .917 .830 504 EXAMPLES. Allowance may be made for the principal hydraulic resist- ances by taking z/ 2 f 2 ~ l as the loss of head before entering the wheel and J72 / 4 2 as the loss of head before entering in the wheel-passages. <5 Then the total loss of head 7/2 Y 2 v 2 =/+/+- ..... (.9) The values of the empirical coefficients^ and/ 4 may vary, the former from .025 to .20, and the latter from .10 to .20. Ex. i. Water enters an O. F. turbine of 3! ft. exterior and if ft. in- terior diameter with a whirling velocity of 20 ft. per second, and leaves in the reverse direction with a whirling velocity of 10 ft. per second. The wheel makes 240 revolutions per minute. Find the useful head. it . 1 1 . 240 Ui = --- = 22 ft. per sec., u% = 2Ui = 44 ft. per sec. Then, if H is the useful head, and the term is 2 - V 2 In order to secure the advantages of centrifugal force, Belanger proposed that the wheel -passages should be so formed that the path of a fluid particle would gradually approach the axis of rotation. Lip (or Tip) Angles. The angles a and /3 which the wheel-blade tips at inlet and outlet make with the wheel's peripheries are generally obtained as follows: From the triangle acd, sin (a -j- y) u l = = = cos y -4- cot a sin y, sin a v, r " and therefore u cot (i 80 a) = cot a =? cot y cosec y. (31) THEORY OF TURBINES. From the triangle fkh, sin (ft + tf) sn = -i = -os tf + cot ft sin and therefore cot (180 ft) = cot ft = cot 6 cosec d. (32) Conditions Governing the Efficiency of Turbines. The whole of the water's energy should, if possible, be employed in doing useful work on the wheel, and the water should there- fore leave the 'wheel without velocity, or v z should be nil. This condition cannot of course be realized in practice, as no water would then pass through the wheel and consequently no work could be done. For purposes of efficiency it is usual to make v z small by adopting one of the following hypotheses: EITHER that the velocity of whirl at outlet is nil, OR that at the outlet the relative velocity of the water and the peripheral linear velocity of the wheel are equal. FIRST consider the hypothesis ' ' that the velocity of whirl at outlet is nil." Then (33) o. I.F. A.F. FIG. 296. FIG. 297. Thus the direction of v is radial in an I. F. or O. F. turbine, Figs. 296 and 297, and vertical in an A. F. turbine, THEORY OF TURBINES. Fig. 298, and therefore the angle hfk (= 6) in the outlet tri- angle of velocities must be a right angle. Hence and z> 2 = v r " = u 2 tan ft K 2 sin /J, . 772 7/ 2 7; 2 ?/ 2 tan 2 fl r 2 "9 " ' " * 2 2 *'**** A^ * Also, eq. (5) gives z/ t sin yA^ = v 2 A 2 = u 2 tan ftA 2 . . General Deductions . (34) (35) (36) IN AN I. F. OR O. F. TURBINE Also, disregarding blade thick- ness, A\ = iTtr^di and y? a = 27Cr*d*. Relation between the lip-angles. By eq. (36) and the triangle acd> jdi sin y u\ sin (a. + y} * tan ft , (37) or 1 ' cot ft = cot y + cot or. . (38) IN AN A. F. TURBINE Also, disregarding blade thick- ness, A, = iitRdi and A* 2 Relation between the lip-angles. By eq. (36) and the triangle acd, d\ sin y _ Ui _ sin (a + y) 2 tan sin a or ~- cot /Sf = cot y + cot a. (38) REACTION TURBINES. IN AN I. F. OR O. F. TURBINE. Speed of turbine. By eqs. (24), (35), (37), IN AN A. F. TURBINE. Speed of turbine. By eqs. (28), (35), (37), tan /? + 2-7- cot (39) Velocity of efflux. = uJ tan tan tan /tf + 2 cot X -. . (40) tan /? + 2- cot (39) Velocity of efflux. vj = u pi, i.e., if / di \ fr^\ 2 sin'M i +2 -cot y cot fi \ > f 1 . If the turbine is drowned with a head h' of water over the outlet, there will be no back-flow of water if pi > pi + oo/i', i.e., if sin 2 y(i +2-^ cot y cot /3 An approximate estimate of the speed of the turbine may now be obtained by making the efficiency perfect, when u^=g(H, +/;). . (45) By eqs. (20), (37), (39), the difference between the inlet and outlet pres- sure-heads = _tv W (46) I \ i /J If the turbine is above the surface of the tail-water, there will be no in- flow of air if pi > pi, i.e., if <, + 2~coty cot/?^ & If the turbine is drowned with a head h' of water over the outlet, there will be no back-flow of water if pi > pi + <*>h', i.e., if sin 2 r{ i + 2-/- cot y cot/5 Speed of turbine. By eqs. (21), (37), ri ri di sin y .- IMPULSE TURBINES. Speed of turbine. By eqs. (21), (37), . (47) Velocity of efflux. Vv=Uv tan /? = -^ i sin . (48) Velocity of efflux. di . V* u* tan p = sin di . (48) THEORY OF TURBINES. Quantity Q of water passing through the turbine per second, blade thick- ness being disregarded. Q = 2rtridiV r ' = 27tridiV 1 sin y = 2itridi sin y V 2 gHi> (49) The useful work (disregarding hy- draulic resistances) (50) (50 r, = coQffJi - The corresponding efficiency Quantity Q of water passing through the turbine per second, blade thick- ness being disregarded. Q = 27tRd l v r ' = iitRdwi sin y = 27tftd l sin y ^gH^ . (49) The useful work (disregarding hy- draulic resistances) The corresponding efficiency. H> d? . An expression can also be easily obtained giving the effi- ciency (eq. 51) of the A. F. turbine independent of the head, H r Thus, by eqs. (28), (35), and (47), ^ sin 2y tan ~" sin y tan 2 h It may be assumed, as a first approximation, that in im- pulse turbines the whole of the water's energy at inlet is transformed into useful work. Then Therefore = 2// T COS y = 2 COS SECOND. Consider the hypothesis " that at the outlet the relative velocity of the water and the peripheral linear velocity of the wheel are equal. " Then .(52) THEORY OF TURBINES. 515 The triangle of velocities, fkJi, at outlet is now therefore an isosceles triangle, in which fk = kh, and the angle hfk = $ = 90 . Therefore v 2 = 2 2 sin - = 2 F 2 sin - (53) Eq. (5), again, gives Aj.\ sin y A 2 V 2 sin ft = A 2 u 2 sin ft. . . (54) O.F. A.F. FIG. 302. FIG. 303. FIG. 304. General Deductions. IN AN I. F. OR O. . TURBINE U\ Ui :==&?. Also, disregarding blade thickness, Relation between the lip angles. By eq. (54) and the triangle acd, Figs. 302, 303, riVi sin y _ i _ sin (a + y) sin sin a or -^rr cosec ft cot x + cot a. (56) IN AN A. F. TURBINE Also, disregarding blade thickness, A l = 27TAW, ; *A* = 27TAW 2 . Relation between the lip angles. By eq. (54) and the triangle acd, Fig- 304, di sin y _ Ui sin ( fa, i.e., if sin 2Y r pi + GO/I', i.e., if H\ h' r-fdi sin ft H > r^~ If the turbine is above the surf-ice of the tail-water, there will be no in- flow of air , i.e., if i + h d* if pi > sin 2y sin ~ft > If the turbine is drowned with a head h' of water over the outlet*. there will be no back-flow of water if /i > / 2 -f- fiV, i.e., if sin iy H l + h > sin IMPULSE TURBINES. IN AN I. F. OR O. F. TURBINE. By eqs. (29), (.52), , = V\. . . . (64) IN AN A. F. TURBINE. By eqs. (30), (52), igh = u~ = 2gH r Then In an /. F. reaction turbine ^=4 = -'. 2 <> y usually varies from 10 to 30, an average value being 20. If u =r K ft usually varies from 135 to 150, an average value being 145. If v^' = o, ft usually varies from 30 to 45, an average value being 35. 5 20 PRACTICAL COEFFICIENTS. In an O. F. reaction turbine v V r " = .21V tO .17^, Let # be the number of the guide-blades. Let //j be the number of the wheel-blades. Then = 4 X shortest distance between wheel-blades, 2,r - = shortest distance between guide-blades, n = -;/! to - r The H. P. = .ijrfH. y usually varies from 20 to 50, an average value being- 25. ft usually varies from 20 to 30, an average value being; 25- In an A . F. reaction turbine % /= v r " =-i$v to ,2v, u^ u. 2 = .567-. Y usually varies from 15 to 50, an average value being 25. .ft usually varies from 12 to 30, an average value being 25. For a delivery of 30 to 60 cu. ft. and a fall of 25 to 40 ft. r =i5toi8 and ft = 13 to 16. For a delivery of 40 to 200 cu. ft. and a fall of 5 to 30 ft. r = 1 8 to 24 and ft = 16 to 24. EXAMPLES. 521 For a delivery of more than 200 cu. ft. and for falls of less than about 5 or 6 ft. y = 24 to 30 and fi = 24 to 28. Denoting \ / A l sin y by A', R may vary from -A' to 2 A' if A' < 2 sq. ft. 2 " " " " -A f to- A if A' > 2 sq. ft. and < 16 sq. ft. i *"* " " " " ^ r to -^' if A' > 200 sq. ft. 4 In A. F. impulse turbines R is often made to vary from -A' to 2^4 '. 4 In reaction and impulse turbines the blade thickness varies from -J to f in. if the blades are of wrought iron, and from % to -f in. if they are of cast iron. The tips of cast-iron blades are usually tapered. Ex. i. An axial-flow impulse turbine passes 170 cu. ft. of water per second under the head of 8.6 ft. over the inlet, and it may be assumed that the whole of this head is transformed into useful work. The depth of the wheel is .9 ft., its mean diameter is 8.4 ft., and the outlet-lip makes an angle of 72 with the vertical. The turbine has 62 guide- and 60 wheel- vanes, all the vanes being ^ in. thick. The outlet velocity of whirl is nil. Find the direction of motion of the water at inlet, the slope of the wheel vane at inlet, the H.P. , the speed, and the inlet and outlet orifice areas and widths. First. Disregard hydraulic resistances. Z/, a //,V W ' UiV- COS Y Then -- = 8.6 = - - -, and Vi = '21/1 cos y = 8 |/8.6 = 23.4606 ft. per sec. AISO, I 7 , 2 = V-? -f i a 27/i//i COS y = 11^ = 7/ 2 a . Therefore V\ u^ = a , and the triangle acd is isosceles, so that a 1 80 iy. 9 f 2 Q /C Again, = the efficiency = i = ' = .905, 2^ x 9.5 2g 9.5 and v-i = 8 \/.$ = 7.58946 ft. per sec. 522 EXAMPLES. Therefore u, = u* = v, cot 18 = 23.358 ft. per sec., 2W, and sec y = = 1.99125, so that y 59 52', and <* = 1 80 2 y 60 16'. The H.P. = 6*JL!Z?JL*5 x i905 = 166 ., 36 . 60 x 23.358 The speed in revolutions per mm. = g = 53.08. The inlet area = ^V = ^ = 8.38 sq. ft. z> r ^j sin y The outlet area = = -^- = 22.4 sq. ft. ^ 7-5 8 9 8.38 = di j TT x 8.4 cosec 59 52' cosec 60 16' > = tt, x 20.53396. and d t = .408 ft. 22.4 = d* < Tt x 8.4 cosec 1 8 \ ~ d t x 18.30983, ( 2 4 ) : and -449 sq. ft. 1 70 1 70 The outlet area = = 5 = 32.05 sq. ft. v-i u-i tan lo 9 , ( 62 60 n , ) 10.449 = **' ) n x 8.4 cosec 47 21' cosec 85 18 > 10 ' 24 24 ) = 18.3402 x di, and d t .57 ft. 9 , \ 60 ) 32.05 = -a* -} TC x 8.4 cosec 18 r J 10 ( 24 ' = d't x 14.85885, and r/ 2 = 2.157 ft. Ex. 2. An A. F. reaction turbine of 7 ft. mean diameter passes 198 cu. ft. of water per second under a total head of 13.5 ft., the depth of the wheel being i ft. At inlet the lip angle (a) is 90, and at outlet the peripheral and relative velocities are equal ( V* = u* = i). The width of the wheel is i ft. at inlet and 1.25 ft. at outlet. Determine the di- rection and magnitude of the velocity of the water at entrance, the up angle at outlet, the speed in revolutions per minute, the efficiency and the H P. Disregard hydraulic resistances. By the condition of continuity, 7T . 7 . I . Vr = 198 = 7C . 7 . ityr", and therefore v r ' = 9 ft. per sec., vr" = j\ ft. per sec. Again, 64 X 13.5 - 7', 2 = 864 V r ' 9 - i* = JV VS = Ui* - V r ' \ or 2i* = 864, or MI 12 4/3 ft. per sec. M 9 , MI 12 4/7 cot y ; = - - = 2.309, and y = 23 25 , Vr 9 sin ft = y- = ~ = -1^ -- = - 4/3 = .3464, and ft = 20 5'. Therefore 8 = ^(180 - 20 5') -.= 79 77^', v* = 2w a sin = 24 4/3 x .1744 = 7.25 ft per sec. The efficiency = i - z = i - .0608 = .9391. 64 x 13.5 The H.P. = 62 * X T ?:! X I98 x .9391 = 285.25, 524 EXAMPLES. 60 x 1 2 f/3 Revolutions per min. = = 56.68. 7t X / Ex. 3. To construct an O. F. turbine from the following data ; the fall (//i)= 5 ft. ; the interior diameter (2^1) = 1.8 ft.; the exterior diameter (2r 2 ) = 2.45 ft. ; Q = 30 cu. ft. per second; y 30 ; the effi- ciency (rf) = .9. Also, disregard hydraulic resistances. First. Take v w " = o. Then v = 4 ^2 ft. per sec., and UiVi = 96 \ 3. Again, by the condition of continuity (eq. 5), TT x i. 80 x d\v\ sin 30 = 30 = n x 2.45 x - /f = a 9.8 r 2 COS -. 7 2 Taking ^ = ^ t .9^/i = 9.8 i^2 cos -. Hence 1^ 4/3" = 9-8 ^2 CQS /? 3 n 2 or. cot | = 432Q j/3 = 3 anc * ^ = 35 34' = outlet-tip angle. Hence t also Vi = 14-6634 ft. per sec., ^ 2 = 9.261 ft. per sec., and //, = 6.7963 ft. per sec j Again, 8in f +jg) ^ = 1.8 ^ '. ^ ^9 sm a 2/j 2.45 zv 2.45 /tf' x _ ^ or cos 30 + cot a sin 30 = - ^ cosec /J = .46399, or cot (180 a) = .804, * nd = 128 48' - inlet-tip angle. 5 26 EXAMPLES. Disregarding the thickness of the vanes, 30 30 54 /3 the inlet area = A, = , = - : - - = .- sec = 4.092 sq. ft.,. Vr Vi Sin 30 9.8 4/2 2 4 092 and d, = n x i 8 = .7233 ft. - d* ; the outlet area = ^- a = -^rr x 1.05018 = 5.5694 sq. ft. t* cos I 2 ^ 60 x 6.796 The number of revolutions per mm. = - = 72. Ex. 4. An I. F. reaction turbine of 24 ins. exterior and 12 ins. interior diameter passes 400 gallons of water per second. The inlet and outlet orifice areas are equal and the depth of the latter is 1.25 ft. The guide- vane lip has a slope of i in 5 and the inlet-lip is radial. Disregarding vane thickness and hydraulic resistances, find the total head over the inlet and also the efficiency, the outlet velocity of whirl being nil. By the condition of continuity, AiVr' = AiVr = 4OO -3- 6i = 64 = A Mr" = AiVt. Therefore V r '- = V r " = V* = 64 -5- 7t . I . I = 8^5 ft. per SCC., and the head equivalent to z/ a = p- = f J = 1.036694 ft. \jj/ Again, Ui = Vr cot y = $v r ' = 4o T 8 7 ft. per sec. = 2 3 , and the useful head = U ^- = = (4O T 8 T > -s- 32 = 51^ ft. * o Hence the /0/d/head = 1.036694 + 51.834710 = 52.871404 ft., and the efficiency = = .98. 7 52.871404 y Also, the speed in revolutions per min. = - X 4C>TT = 388.76. The H.P. = It X 2 62 * X 64 x 55 ^ tan /? = = 8 F 8 5 -5- 2o T 4 r = .4, and /5 21 48'. EXAMPLES. 527 Ex. 5. In the preceding example show how the results will be mod- ified if, instead of the outlet velocity of whirl being nil, the relative and peripheral velocities at outlet are equal. As before, v r ' = v r "= 8 5 8 - ft. per sec., ft. per sec. 2 2 = 2 1/2". The speed in revolutions per min. = - U = 388.76. /t X 2 Again, V* - V? = , - ,* + Or 2 J 2/ r ' a = Wa' j* + and H, , the total head, = o Also, sin = -r- = - = 8& -8- 20 T 4 T = .4, and ft = 230 35'. The effiaency -,_ '' ^ = 32 x 51.834710 The H.P. = ^_i^r2_i^i!r^ii x .979 = 369.109- Ex. 6. Avortex impulse turbine, without guide-vanes but with 32 wheel- vanes of f-in. thickness, has an exterior diameter of 2.625 ft., an interior diameter of 2.1 ft., and passes 30 cu. ft. of water per second under a head of 560 ft. The water enters at an angle of 30 with the wheel's periph- ery, and the relative and peripheral velocities at outlet are equal. The wheel depth at outlet is 3 times the depth at inlet. Allowance is made for hydraulic resistances by taking .94 as a coefficient of velocity at inlet, and by adding 10 per cent to the head equivalent to the relative velocity at outlet. T/J = .94 1/64.560 = 177-955 ft. per sec. 1 1 Also, U-? u\ Fy 10 _ a } i 11 10 . /25o HI sin ( ~ cosec 30= = -337941 sq. ft., and di = .06 ft. = . Also, . ( 1 , ) 30 cosec 10 .I8-J7TX2.I 32 x - cosec ft [ Ai = ( 48 | , 3 /? - - cosec p, 80.5072 or 1.0692 = cosec fi (.324 + .372637) = cosec ft x .696637, and cosec ft = 1.534, or ft = 40 41'. Therefore, also, 8 = 69 39^'. Again, v w ' = 177.955 cos 3 = I 54- II 3 ^. P er sec., and W = 2 (i cos ft) = 80.5072 x .241676 19.4723 ft. per sec. Hence the efficiency = loa6 * X 154 "' ' 3 " 8 ' 5 72 X ' 4y ' } 32 x 560 = '3941 . ?3 17920 The H.P. = DRAFT-TUBES. 529 7. Theory of the Suction (or Draft) Tube. Vortex and axial-flow turbines sometimes have their outlet-orifices opening into a suction (or draft) tube which extends downwards and discharges below the surface of the tail-water. By such an arrangement the turbine can be placed at any convenient height above the tail- water and thus becomes easily accessible, while at the same time a shorter length of shafting will suffice. The suction tube is usually cylindrical and of constant diameter, so that there is an abrupt change of section at the outlet-surface of the turbine, producing a corresponding loss of energy by eddies, etc. This loss may be prevented by so forming the tube at the upper end that there is no abrupt change of section, and by gradually increasing the diameter downwards. The cost of construction is greater, but the action of the tube is much improved. Let h' be the head above the inlet-orifices of the wheel. Let h" be the head between the inlet-orifices and the sur- face of the tail-water. Let L^ be the loss of head up to the inlet-surface. Let L 2 be the loss of head between the wheel and the tube- outlet. Let z/ 4 be the velocity of discharge from the outlet at bottom of tube. Let P be the atmospheric pressure. Then, assuming that there is no sudden change of section at the outlet-surface, W and therefore P V 2 V 2 JL _l_ 12 _ 1 4 I W ^ ~ ~^ 53 DRAFT-TUBES. where // = h' -\~ h" = total head above tail-water surface, and 7' 2 2 , v, L l , L 2 are expressed in the forms < I , /* 4 , /i 5 , // 6 being empirical coefficients. Again, the effective head . 2 and is entirely independent of the position of the turbine in the tube. Also, if A i is the area of the outlet from the suction-tube, ^4 7 4= Q ~ A < i \ sin Yi so that 7' can be expressed in terms of v., and hence - 2 ze/ is also independent of the position of the turbine in the tube. Suppose the velocity of flow to be so small that z/ 4 , v 2 , L. 2 may be each taken equal to nil. Then *"+* = ; W W and since the minimum value of p 2 is also nil, the maximum- theoretical height of the wheel above the tail-water surface is equal to the .head due to one atmosphere. Again, g( h> + //') = gH =-- r.X - Vw "u,+ V - v 2 = ^ cos y Ul - u 2 (u 2 V 2 cos fl) + -. But AJL\ sin 7 Q = A 2 v 2 sin 6 = A 2 V 2 sin /? = Av\ and hence, taking cos . u cos ? 2 -- LOSSES AND MECHANICAL EFFECT. 53* and therefore cos y 4- v w . cos where B = f cos y -\- y// 3 cos p\ Hence it follows that v l increases with u 2 , i.e., with the speed of the turbine, if A suction-tube is not used with an outward-flow turbine, but a similar result is obtained by adding a surrounding sta- tionary casing with bell-mouth outlet. A similar diffusor might be added with effect to a Jonval turbine working without a suction-tube below the tail-water. The theory of the diffusor is similar to that of the suction-tube. 8. Losses and Mechanical Effect. The losses may be enumerated as follows: I. The loss (Z,j) of head in the channel by which the water is taken to the turbine. L -f 1 -^ 1 ~ /1 m2g' _f l being the coefficient of friction with an average value of .0067, / the length of the channel of approach, m its mean hydraulic depth, and z' the mean velocity in the channel. L l is generally inappreciable in the case of turbines of the inward- and axial-flow types, as they are usually supplied with water from a large reservoir in which ?' is sensibly nil. If A Q is the sectional area of the supply-channel, then - where / 4 varies from . 10 to .20. NOTE. The loss of head due to skin-friction often governs the dimensions of a turbine, and renders it advisable, in the case of high falls, to employ small high-speed turbines. V. The loss of head (Z 5 ) due to the abrupt change of sec- tion between the outlet-surface and the suction-tube. As in III, v 2 (=f/i) is suddenly changed into v% (= fh')> and the loss of head is 2g 2g since h ' x is very small and may be disre- Uz garded. Thus FlG - 3<*. *^ vj being the component of v^ (fh f ) in the direction of the axis of the suction-tube. If there is no abrupt change of section between the outlet- surface and the tube, Z 5 is nil. VI. The loss of head (Z, 6 ) due to friction in the suction- tube. Assume that the velocity v^ of flow in the tube is equal 534 LOSSES AND MECHANICAL EFFECT. to v 2 ', the velocity with which the water leaves the turbine. Also let A be the sectional area of the tube. Then o f & (=f^ being the coefficient of friction with an average value of .0067, /' the length of the tube, and m its mean hydraulic depth. VII. The loss (Z- 7 ) of head due to entrance to sluice at base of tube. This loss may be expressed in the form 7 ,EL ~~ the average value of/ 7 being about .03. VIII. The loss (Z- 8 ) of head due to the energy carried away by the water on leaving the suction-tube. and v^ usually varies from \ V^gH to f \/2gH. In good turbines the loss should not exceed 6 per cent. It might be reduced to 3 per cent, or even to I per cent, but this would largely increase the skin-friction. IX. The loss of head (X 9 ) produced by the friction of the bearings. fj. being the coefficient of journal friction, Wthe weight of the turbine and of the water it contains, and p the radius of the journal. Hence the total loss of head = L 1 + L i + L 3 + L t + L l + L t + L r + L s + L, = L, and the total mechanical effect EXAMPLE. 535 NOTE. If there is no suction-tube, L 5 = o = L 6 = L^ = Z g , and the total loss becomes r , r T r T \ v ? j fal1 from outlet-surface to ^i -f ^ 2 -f ^3 4 ^4 , A, + ^ H- { tail-water surface. EXAMPLE. A vortex turbine, with a draft-tube of the same sectional area as that of the outlet-orifice openings, passes 100 cu. ft. of water per second under the head of 9^ ft. The exterior and interior diameters are in the ratio of 5 to 4, and the outlet- and inlet-areas are in the ratio of 9 to 10. The direction of the water at the inlet and the outlet lip angle are given by sin y = .25 = sin ft. The water leaves the tube through a sluice having a sectional area 10 per cent greater than that of the outlet-orifice area. The outlet velocity of whirl is nil, i.e., d =90. Disregard the losses L l , Z 3 , L 5 , L 6 , and L g . v 2 The loss of head to inlet =f 2 --. V* 44 " " " in wheel-passages = f^2-. o V 2 " " " " at sluice-entrance =/-*-. " " " " carried away by water = . Hence the total loss of head But, by the condition of continuity, ^^ sin r = g = ^ 2 F 2 sin ft = Aft. Therefore Vl _A 2 sin^_ ^4_^_ 2sil ._^5 T^^T/STJ;- v~ A* 1.1 S3 6 EXAMPLE, Hence the total loss of head taking / 2 = .1, / 4 = .126, and / 7 = .03. Again, the "useful" head = = - - . .#,.#, cos v 32 32 4 5 ^4 2 F 2 sin/? cos v F 2 135 = -~ PI COS /? . -V^ . ~ = - . -/^- = 2.109375 ft. 128 2 A i smy 2g 64 Therefore 9.5 = (.26 + 2.1094) = ~ ~ X 2.3694, y j s* ,_\ - 7 ~/ /- n- ** s^' or F 2 ^ = 4 ft., approx., and F 2 = 16 ft. per sec. The useful work per Ib. of water = 4X^ = 8.4375 ft.-lbs. The work consumed in hydraulic resistances per Ib. = 4 X .26 = 1.04 ft.-lbs. The total work per Ib. of water = 9-4775- The = ^? C osec/S=^ = 25 sq. ft. and ^, = = 27.78 sq. ft. EXAMPLE. Again, v \ ~ -9^2 = J 4-4 ft. per sec., 2 = F 2 cos/? = l6 V^= 1 5-492 ft. per sec., and ft- per sec. Also, u l sin (<* -|- y) or U l ~ = ^ ( -f~ X4- 3-8/3= 1.5061. and =33 35'- If the diameter of the tube is equal to that of the outlet- surface, viz., 4 ft., and if its lower edge is rounded so that / 7 = O, then energy per Ib. of water carried awav = ~^- _ V} . 015448. The loss in shock in draft-tube F,' i / I75^ 2 F/ : ^?6 I 1 "^ == ^ X - Thus the total loss now becomes F 2 772 2^-081 + .126 + .05448 + .04705) = ~X .2695. V* As before, the useful head = 2 - x 2. 1094. o 53 8 EXAMPLE. V* Therefore the total head = ~^- X 2.3789, 2. 1094 and the efficiency = 2 g = Also, o c = - 2 -X2.378o, or -*-= 3.993 ft., and V 2 = 15. 987ft. persec. 2- 2- v 2 If there is no draft-tube, must be substituted for - * vj_ __ V? JL Thus the total loss of head is now L2.(.o8i + .126 + .0625) = -^ X -2695, F 2 which exceeds the loss of head with a draft-tube by X .0095 = .038 ft., which is less than four hundredths of a foot and is practically inappreciable. EXAMPLES. 539 EXAMPLES. 1. A downward-flow turbine of 24 ins. internal diameter passes 10 cu. ft. of water per second under a head of 31 ft.; the depth of the wheel is i ft. and its width 6 ins. Find the efficiency, assuming the whirling velocity at outlet to be nil. Ans. .997. 2. A downward-flow turbine of 5 ft. external diameter passes 20 cu. ft. of water per second under a head of 4 ft., the depth of the wheel being i ft. The water enters the wheel at an angle at 60 with the ver- tical, the receiving-lip of the wheel-vanes is vertical, and the velocity of whirl at outlet is nil. Find the internal diameter and the speed in rev- olutions per minute. Ans. 4.6 ft.; 46.53. 3. A downward-flow turbine has an internal diameter of 24 ins. ; the breadth of the wheel is 6 ins. ; the turbine passes 33 cu. ft. per second under an effective head of 16 ft. Assuming the whirling velocity at out- let to be nil, find the efficiency and power of the turbine. If the vane- lip at inlet is vertical, find the direction of the vane at outlet, and the speed of the turbine in revolutions per minute. Ans. .931 ; 55.865 H.P. ; = y =21 2'^ 166.7. 4. Discuss the preceding example on the assumption that the pe- ripheral speed at outlet (w a ) is equal to the speed of the water at that point relatively to the wheel ( F 2 ). Ans. .928 ; 55.715 H.P. ; ft = 21 47' and y = 20 21'. 5. An axial-flow impulse turbine of 5 ft. mean diameter passes 170 cu. ft. of water per second under an effective head of 8.6 ft. ; the depth of the wheel is .9 ft. At what angle should the water enter the wheel to give an efficiency of 81 per cent, the width of the wheel being constant and disregarding hydraulic resistances? z/ w "=o. Ans. = 27 16'. Also find () the velocity with which the water enters the wheel; (ft) the speed of the turbine in revolutions per minute; (c) the directions of the vane-edges at inlet and outlet ; (d) the velocity of the water as it leaves the wheel ; (e) the power of the turbine. Ans. (a) 23.46 ft. per second ; (6) 45.08 ; (c) a 130 10'; ft = 42 19' ; (d) 10.748 ft. per second ; (e) 148.65 H.P. If, instead of assuming that the whirling velocity at exit is nil, it is assumed that the peripheral speed (,) of the wheel at the mean radius is equal to the relative velocity ( F a ) of the water at exit, show how the several results are affected. Ans. Y = 25 8' ; (a) 23.46 ft. per second : (b) 54.638 ; (c) a = 124 49', ft = 44 6'; (d) 10.748 ft. per second ; (e) 148.65 H.P. 540 EXAMPLES. Also show how the results are affected when it is assumed that the hydraulic resistances necessitate an increase of 12^ per cent in the head equivalent to the velocity with which the water enters the wheel, and an increase of 10 per cent in the head equivalent to the relative velocity ( F 8 ) at outlet. Ans. When i / w " = o (a) 22.12 ft. per second; () 44.21; (c) a = 147 50', /3 = 27 44; (d) 10.748 ft. per second ; (e) 148.65 H.P. When Ui = V* (a) 22.119 ft P er second ; (b) 50.97; (c) a= 123 19', ft = 47 28' ; (d) 10.748 ft. per second ; (*) 148.65 H.P. If the turbine has 65 guide-blades of .2-in. thickness and 63 wheel- vanes of .4~in. thickness, find the widths of the inlet and outlet openings. Ans. If v w " = o, d\ 4.214 ft., dt = 2.83 ft. If u, = F 3 , d, = 1.78 ft., d, = 1.48 ft. 6. The efficiency of an axial-flow turbine of 4 ft. mean diameter is 90 per cent, and it passes 12 cu. ft. per second under an effective head of 40 ft. At the mean radius the water enters at an angle of 30 with the wheel's face, and the whirling velocity at outlet is nil. Find (a) the velocity with which the water enters and leaves the wheel ; () the directions of the vane .at inlet and outlet ; (c) the sectional areas of the inlet- and outlet-orifices ; (d) the speed of the wheel in revolutions per minute ; ( the speed of the wheel in revolutions per minute; (* Q } <*>* co } I 2nr where 2nrd Y = (2nr nt)d', T= work of pump to radius r\ r i = radius to inner end of blade; v r = radial velocity at inlet; n = number of blades ; t = thickness of blade. By plotting the values of corresponding to different values of r the curve of the blade may be defined. It is essential that there should be no dissipation of energy in eddy motion at the inlet, and the direction of the relative velocity, V l , should therefore be tangential to the blade-tip at a, Fig. 315. Then, from the triangle adc, V* = v* -)- u^ 2v l u l cos y, (5) ' fkh, V* = v* + u? - 2v ^ cos tf, (6) and therefore V? - - F 2 2 z/ 2 2 u? \y* v* _ v 2 u 2 cos d T/ I ?/ I cos y . ig *g ~^g~ g ~ir 7) The water leaves the wheel with a velocity v 2 , and carries v 2 away ; in its energy of motion, viz., , an important; portion of the work done on the pump by the prime mover. If the whole of this energy could be made available for increasing the pumping power, then, by Bernoulli's theorem, i ( . 2g W 2g W 2g Also, y 2 __ U 2 the term 2 L being the variation of pressure-head due to 55 8 centrifugal action between the \vheel inlet and outlet. Hence,, by eqs. (8) and (9), 7, 2 T72 772 ?/ 2 _ 2 7 , 2 _ 7 , 2 A r + ff,+ -= ^ - - 2 + ^- -'- + -'- - L a ' ? cr -7 (v " 1' ) V U COS $ 7^ // COS V V 2 a u ~~ v vu cos and therefore II * \/ * TT I IT TT \ T7 ,* XT' 11 /^/\O ir TT U 2 V 2 I V *V V ^ V s) V * V 1 U 1 COS 7 X , N H = ^- ~Y~ '^~' K ~ t (I4) r., 2 / , / Maximum Efficiency. If the terms h r , , and - J ^- !L 6 " o MAXIMUM EFFICIENCY OF CENTRIFUGAL PUMP, 561 are sufficiently small, as compared with H a , to be disregarded without much error, and if y = 90, then eq. (14) becomes Ha = ^^ (15) But sin tf sin cos 6 = v a cos 8 = 9 9 /JQ , Jv * w 2 2 /ft I ' 2 sin (ft + tf) 2 sin (/? + rf) Therefore sin 2 2 sin/?sin(/? + 2d) ,gH a ~u*\\ -T- "" \^~r u j ' 3111 ~ {r ~r u ; (16) sm 8 (/?+<5)J sin 2 or 2gH a ffi . n and the efficiency ?; = -77 = - 2 cos tf.sin or 77 = -{i + tan tf cot (/?+ rf)[. The efficiency increases as /?, the outlet-tip angle, dimin- ishes, and would be unity, i.e., perfect, if ft could be O. If the blade is radial at the outlet, i.e., if fi 90, then the efficiency -J I + tan 6 cot (90 + 326> But since the angle in a semi- circle is a right angle, DO AD DO tan $ cot (04- 0} - and the efficiency is therefore greatest when is a maximum. is nil both when D coincides with A and also with DB C, and must consequently be a maximum, or stationary, at some position of Z> between A and C. This position is at once found from the condition that if D l is a consecutive point and if D^B is joined intersecting A C in O l , then _ 'DB ~~~ T\B' so that DD l must be parallel to OO l or ^ C, and is therefore a tangent to the semicircle at D y which is necessarily the middle point of the arc AC. Hence, since the arc AD the arc CD, the angle ABD = the angle and therefore 90 - (ft + 6) = <* or /? + 2d- 9 o. MAXIMUM EFFICIENCY OF CENTRIFUGAL PUMP. 563 Hence, too, the max. efficiency I -\- tan tf cot (90 6) = 1 ** = sec* ( 4S - - The outlet velocities corresponding to this maximum effi- ciency are represented by the sides of the triangle fkh, Fig. 327. The two triangles fnh and fxh are equal in every respect. Also, FIG. 327. 2 sin 2 TO " = . EXAMPLES. 575 Also find the proper sectional area of the chamber surrounding the fan for the proposed delivery and lift. Examine the working of the pump at a lift of 15 ft. (v w " = o). Ans. Breadth, $ in. ; revolutions, 700 ; 23.5 sq. ins. 7. For a given discharge (0 and head (//), and considering only the losses of head due to flow and to the resistance in the wheel, show that the maximum efficiency of a centrifugal pump of diameter D is i -A Q ' A being a constant depending on the size of the wheel. 8. A centrifugal pump with an efficiency of .75 and a radial flow at inlet, lifts 35 cu. ft of water per second a height of 20 ft. At the outer periphery the vane-angle (ft) is 15 and the radial velocity is 5 ft. per second. If the wheel makes 140 revolutions per minute, find (a) its diameter. If the diameter of the outer periphery of the wheel is three times that of the inner periphery and if the radial velocity at the latter is 8 ft. per second, find (b) the vane-angle at the inner periphery and (c) the depths of the wheel at the inner and outer peripheries. Ans. (a) 5.455 ft. ; (b) 30 58' ; (c) .765 ft. ; .41 ft. 9. The pump in the preceding example is supplied with a vortex- chamber of 6 ft. diameter. Show that the "gain of head " is a maxi- mum when the velocity of flow in the volute is 8.46 ft. per second. Also show that the frictional loss of head is 4.18575 ft. 10. In a centrifugal pump the diameter of the fan = 12 ins., the depth = 2 ins., the lift = 25 ft., and the delivery = 300 cu. ft. per minute. Determine (a} the speed ; (ff) the efficiency ; and (c) the power expended when the vane-angle (ft) at the outer periphery is (i) 90 ; (2) 45 ; and (3) 30 ; Y b eing 90. Ans. (i) (a) 785 revols. per min. ; (b) .47 ; (c) 30 H.P. ; (2) (a) 805.8 (6) .58 ; (c) 24.4 H.P. ; (3) (a) 846.1 " " " (b) .68 ; (c) 22.9 H.P. 11. A centrifugal pump delivers 10,000 gallons per minute. The actual lift is 50 ft. The radial velocity at the outlet-surface is one eighth of that due to the actual lift and u t = 2v w ". Find (a) the radius of the wheel ; (b) the vane-angles ; (c) the speed of the wheel ; (d) the effi- ciency, taking y = 90 ; and d*. = d* -. Ans. (a) 1.9 ft.; (b) 56 16'; 23 16'; (c) 331 revols per min. ; (d) .74. 12. The internal and external diameters of the fan of a centrifugal pump with radial flow at inlet are 9 ins. and 18 ins., respectively; the depth is 6 ins., and it passes 400 cu. ft. per minute against a pressure head of 16 ft. The inclination (ft) of the discharging-lips of the fan being 30, determine (a) the speed ; (b) the efficiency ; (c) the power ex- 574 EXAMPLES. pendecl ; and (^H a . eh Ans, .73 A and .79 A where A = &-, . 503.6 21. In a centrifugal pump the diameters of the suction- and dis- charge-pipes = 48 ins.; the number of wheel-blades = 6; the blade thickness = $ in.; the radial velocity of flow at outlet = 2.877 ft. per second ; the velocity of flow in the volute and discharge-pipe 5.817 ft. per second; the peripheral speed of the wheel outlet-surface = 34.6276 ft. per second. Disregarding the frictional losses in the suction- and discharge-pipes and in the wheel-passages, determine the velocity of whirl at outlet, the blade-tip angles at outlet, the delivery in cubic feet per second, the speed in revolutions per minute and the actual lift, the efficiency being .759. Ans. 23.695 ft. per second ; ft = 14 44' ; 73.13 cu. ft. ; 80.13 : 19.34 ft. 22. A centrifugal pump, with an actual lift of 10 ft., delivers 37.85 cu. ft. of water per second at a speed of 68 revolutions per minute. The number of blades = 6 ; the blade-thickness = in.; the wheel-depth at outlet = 9 ins. ; the diameters of the suction- and discharge-pipes = 36 ins.; the diameter of the wheel = 90 ins. ; = 19 7' 26.67" ; Y = 90. Find the gain of head in passing from the wheel into the volute and the frictional loss (h^) in the discharge- and suction-pipes and in the wheel- passages. Also find the efficiency. Ans. 2.193 ft- ; 1.911 ft.; .65. 23. In the centrifugal pumps for two torpedo-boat destroyers the diameter of eye = 7 ins. ; the diameter of wheel = 20 ins. ; the number of blades = 6 ; the thickness of blades = T \ in.; the width of the wheel at outlet = !-& ins.; the actual lift = 63^ ins.; cot ft = 5.167. The pumps are driven by a vertical non-condensing engine with a 4|-in. cylinder, a 4-111. stroke, and a $-in. piston-rod. With a boiler-pressure 57 EXAMPLES. of 220 Ibs. per square inch above the atmosphere and a cut-off at f, the delivery was found to be 1113 gallons (U. S.) at 420 revolutions per minute. The frictional losses, due to one upper bend, two 7-111. bends, one- bad check-valve, one gate-valve, and about 8 ft. of 7-in. pipe, were respectively estimated at .3/14, .^hd, /id, .i/id, and .4185 ft., /id being the head corresponding to the velocity of discharge (= velocity of flow in volute). Find (a) the mechanical efficiency; and also find, on the ordinary hypotheses and assuming y = 90, (b) the radial velocity of flow ; (c) the loss in shock on entering the volute ; (d) the hydraulic efficiency. Ans. (a) 6.02 per cent ; (b) 4.014 ft. per sec. ; (c) 61.7 ft.-lbs. ; (d) .434. 24. Show how the results in the preceding example will be affected with a delivery of 2000 U, S. gallons at an assumed speed of 700 revo- lutions per minute. Ans. (a] 6.67 per cent ; (b) 7.214 ft. per sec. ; (c) 120 ft.-lbs. ; (d) .409. 25. Determine the hypothetically best speeds in revolutions per minute for the pumps in Examples 23 and 24, and calculate the corre- sponding maximum hydraulic efficiencies. Ans. In Ex. 23 best speed = 292.7 rev. per min. " " 24 " " = 526 " " " 26. A centrifugal pump delivers 20 cu. ft. of water per second at a speed of 225 revolutions per minute ; the diameter of the discharge-pipe is 20 ins., the diameter of the wheel is 36 ins.; the width of the wheel at outlet is $$ ins.; the number of blades = 6; the blade thickness = f in.; ^ = 90; cosec ft 1.484. Find the hydraulic efficiency, and also find the diameter of the whirlpool-chamber which will increase this efficiency by .1234. Ans. .708 ^- ; 48 ins. 27. A centrifugal pump making 229^ revolutions per minute delivers 23^ cu. ft. of water per second. The diameter of the discharge-pipe = 18 ins., of the wheel = 42 ins., and of its whirlpool-chamber =48 ins. The width of the wheel at outlet = 3.452 ins., and of the whirlpool-cham- ber at its outer circumference = 2.5 ins. The tip angle ft at outlet = cot~ J 3.6. Assuming the ordinary whirlpool theory and disregarding hydraulic resistances, determine (a) the radial velocity of flow (vr")\ (b) the actual velocity, z/ 2 , with which the water leaves the wheel; (c} the loss in entering the whirlpool-chamber ; (d} the hydraulic efficiency. There are six blades each f in. thick. Ans. (a] 9.3569 ft. per sec.; (b) 12.567 ft. per sec.; (c) 76.4142 ft.-lbs.; (d) .49. INDEX. Abbot, 252, 253, 257 Abrupt changes of section, loss of head due to, 164 Accumulators, 339 Accumulator, Brown's steam, 344 differential, 342 Air in a pipe, 183 Air, retarding effect of, 224 Applications of Bernouilli's Theo- rem, 12 Aqueducts, circular, 242 egg-shaped, 244 flow-in, 240 square, 243 Arc of discharge in overshot wheel, 452 Aspirator, 16 Axial-flow turbine, 490 Balancing of hoists, 345 Barker's mill, 375 Barlow's curve, 73 Barnes, 130 Barometer, water, 7 Bazin, 230, 246, 248, 249, 250, 252, 257, 258, 260, 266 Bazin's velocity curve and formula, 265, 266 Bazin's weirs, 99 Bear, punching, 339 Beardmore, 247 Beaufoy, 122 Belgrand, 226 Belgrand's sewer formula, 246 Belidor, 386 Bellmouth, 36 Bends in pipe, 168 Bends, river, 269 Bernouilli's Theorem, 8 applications of, 12 Bidone, 60, 284 Binding-press, 338 Boileau, 268 Boileau's velocity curve and for- mula, 268 Borda, 60 Borda's mouthpiece, 58 Borda's turbine, 382 Bordered vane, 368 Bossut, 418 Bourgogne canal, experiments on, 249. 257 Bovey's tables of coefficients of dis- charge, 39, 40 Boyden's hook gauge, 298 Boyden's diffusor, 492 Brakes, hydraulic, 353 Bramah's press, 336 Branched pipe connecting three reservoirs, 191 Branch main of uniform diameter, 188 Breadth of water-wheels, 438 Breast-wheels, 440 Breast-wheel, efficiency of, 441 losses of effect in, 442 mechanical effect of, 442 speed of, 441 Bresse, 291, 292, 296, 309 Broad-crested weir, 94 Brotherhood hydraulic engine, 345. Brown's steam-accumulator, 344 Brumings, 247 Bucket, capacity of, 458 form of, 435, 458 Buckets, number of, 458 Burdin's wheel, 385 Canal-lock, time of emptying and filling a; 59 Capacity of water-wheel buckets, 458 577 578 INDEX. Capillary phenomenon, 130 Capillary tubes, flow in, 130 Castel's table of mouthpiece co- efficients, 69 Centre of pressure, xiv Centrifugal force, effect of, 451 Centrifugal pump, 76 analysis of, 553 efficiency of, 555 height of suction in, 554 losses due to hydraulic resistance in, 554 values of a, ft, and y in, 556 vortex chamber in, 565 work of, 394, 555 Centrifugal turbine, 393 Chamber, whirlpool, 76, 565 Channel-flow assumptions, 220 Channel, bottom velocity of flow in a, 266 flow between bridge piers in a, 296 flow in an open, 221 flow through contracted portion of a, 293 form of, 228 maximum velocity of flow in a, 236, 258, 265 mean velocity of flow in a, 268 mid-depth velocity of flow in a, 265 of great width as compared with the depth, 288 of rectangular section and small slope, 287 steady flow in a, 221 surface velocity of flow in a, 265 value of tfand ft in a, 249; of y in a, 250; of n in a, 251 variation of velocity in a section of a, 257 Channels, cycloidal, 239 differential equation of flow in, 275 examples of, 228 longitudinalprofile of, 285 of constant section, steady flow in, 271 of varying section, flow in, 271 rectangular, 229 semi-circular, 238 semi-elliptic, 239 surface-slope in, 227 trapezoidal, 231 with change of section, 293 with constant mean velocity of flow, 235 Chezy's formula, 163 Chezy's experiments on Courparlet channel, 247 Circular orifices, 81 Cock in cylindrical pipe, 169 Cocks, loss of head due to, 169 Coefficients, hydraulic, 29 Coefficients for turbines, 519 Coefficient of contraction, 34 discharge, 38 friction, 124 resistance, 34 velocity, 30 viscosity, 269 Coker, 130 Combined-flow turbines, 495 Compressibility, 25 Constants, useful, xvii Continuity, 27 Contraction, imperfect, 34 incomplete, 35 loss of head due to abrupt, 165 Coulomb, 122 Courparlet channel, experiments on, 247 Critical velocity, 129 Cunningham, 257 Current-meters, 306 Cylinders, thickness of, 337, 344 Cylindrical body in pipe, pressure on, 406 Cylindrical mouthpiece, 63 Danaides, 386 Darcy, 126, 139, 249, 260. 303 Darcy gauge, 302 D'Aubuisson, 126 Defontaine's velocity-curve formula, 262 Density, 2 Diagrams of pipe-flow experiments, 146 Didion, 403 Differential accumulator, 342 equation of steady vaned motion, 275 Diffusor, Boyden's, 492 Divergent mouthpiece, 66 Downward-flow turbine, 490, 494 Draft-tube, theory of, 529 Drummond on Miner's Inch, 44 Dubiat, 247, 258, 403 Dupuit, 293 Efficiency of centrifugal pumps, 555 Efficiency of turbines, conditions governing, 510 remarks on, 519 effect of centrifugal force on, 508 INDEX. 579 Elasticity of volume, 6 Elbows, loss of head due to, 167 Ellis, 177 Energy, losses of energy in hy- draulic machines, 351 lost in shock, 55 of jet of water, 69 of water-fall, 7 transmission of, 156 of fluid, kinetic, n; pressure, n; weight, II Engine, hydraulic, 347 speed of steady motion in, 351 Enlargement of section, loss of head due to, 167 Equations, general, 53 Equipotential surface, 20 Equivalent uniform main, 186 Erosion caused by watercourses, 227 effect of, 226, 227 table of, 269 Examples, 109, 210, 328,355,408,539, 572 Exner. 308 Expansion, cubical, 6 Experimental tank, 29 Eytelwein, 247, 248 Farmer, 49, 81 Flamant, 144 Float adjustment in experimental tank, 41 Floats, sub surface, 300 surface, 300 twin, 301 Flow from vessel in motion, 26 in a frictionless pipe, 27 in aqueducts, 240 influence of pipe's inclination and position upon the, 138 in pipes, 133 in pipe of uniform section, 133 varying diameter, 184 Fluid, definition of, xiii friction, 121 motion, i pressure, xiii rotation, 17 whirling of, 19 Foss, 143 Fourneyron's turbine, 491 Fournie, 142 Francis, 86, 89, 301 Freeman, 178 Free surface, 20 Friction, coefficient of. 124 in pipes, surface, 125 Friction, laws of fluid, 123 Frictionless pipe, flow in, 27 Froude, 131 Froude's table of frictional resist- ances, 121 Fteley, 89 Funk, 247 Ganguillet & Kutter's formula, 250 Gas, definition of, xiii Ganges, experiments on, 257, 278 Gauckler, 253 Garonne, experiments on, 266 Gauge, Darcy, 303 Hook, 298 Gauging, methods of, 297 of pipe-flow, 207 Gaugings on the Ganges, 278; Mis- sissippi, 253 General equations, 53 Gerstner's formula, 421 Graphical representation of losses of head, 170 Grashof, 431 Grassi, 6 Hagen, 139, 253 Head, 27 Hele Shaw, 129 Herschel, 208 Hoists, hydraulic freight, 345 Hook-gauge, Boyden's, 298 Humphreys, 252, 253, 257 Hurdy-gurdy, 485 Hydraulic coefficients, 29 engine, 344; analysis of, 347 gradient, 13 intensifier, 342 jack, 338 N mean depth, 222 mean radius, 135 press, 335 ram, 334 Hydraulic transmission, 156 Hydraulics, definition of, i Hydrodynamometer, Perrodil's, 308 Hydrometric pendulum, 308 Hydrostatics, fundamental prin- ciples of, xiv Ice, weight of, 3 Impact, 359 apparatus, 369 coefficient of, 371 on a flat vane, 359 on a curved vane, 388 on a hemispherical vane, 367 on a surface of revolution, 364 INDEX. Impact on a vane with borders, 368 Impact on a wheel, 378 Imperfect contraction, 34 Inclination, influence of pipe's, 138 Injector, 15 Intensifier, 341 Inversion of the jet, 48 Inverted siphon, 182 Inward-flow turbine, 490, 493 Jack, hydraulic, 338 Jackson, 251 Jet, energy of, 69 inversion of, 48 measurer, 37 momentum of, 69 propeller, 373 Jet reaction wheel, 375; efficiency of, 376; useful effect of, 376 Jet turbine, 400 Knibbs, 142 Kutter, 142, 230, 253 Laminar motion, 2 Lampe, 143 Lesbros, 48 Level surface, 20 Levy, 143 Lift, balanced ram, 345 hydraulic ram, 346 Limit turbine, 494 Lines of force, 20 Liquid, definition of, xiii Lock, time of filling a, 50 Longitudinal profile of open chan- nel, 285 Loss of energy in shock, 55 Loss of head due to abrupt change of section, 164 ; bends. 168 ; cocks, 169 ; contraction of sec- tion, 169 ; elbows, 167 ; enlarge- j ment of section, 167 : orifice in diaphragm, 166 ; sluices, 169 ; valves, 169 Losses of head, graphical repre- sentation of, 170 Losses in centrifugal pumps, 559 in turbines, 531 Magnus, 48 Main, equivalent uniform, 186 of uniform diameter, branch, 188 with several branches, 201 Manning, 230, 252 Mariotte, 403, Metacentre, xv Meters, 207 inferential, 209 piston, 209 rotary, 209 Schonheyder's, 208 Venturi, 207 Meyer, 269 Miner's Inch, 44 Mississippi, experiments on, 253, 267 Mixed-flow turbines, 495 Momentum of jet, 69 Morin, 403, 431 Motion, fluid, I in plane layers, 2 in stream-lines, 2 laminar, 2 permanent, i steady, I Motor driven by water flowing along a pipe, 179 Mouthpiece, Borda's, 58 convergent, 66 cylindrical, 63 divergent, 66 ring-nozzle, 61 Navier's hypothesis, 203, 264 Notch, 83 rectangular, 83 triangular, 92 Nozzles, 174 Ellis' experiments on, 177 Freeman's experiments on, 178 Open channels, 220 Orifice fed by two reservoirs, 195 flow through an, 23 in a diaphragm, loss of head due to, 166 in a thin plate, 22 in vertical plane surfaces, 78 in vessel in motion, 26 with a sharp edge, 22 Orifices, circular, 81 large, 78 rectangular, 78 semi-circular, 49 triangular, 92 Orleans canal, experiments on, 247 Outward-flow turbine, 491 Overshot-wheel, 450 arc of discharge in, 452 bucket angle of, 456 division angle in, 456 effect of centrifugal force in, 451 ; impact on, 469; weight on, 467 INDEX. Overshot-wheel, 450 number of buckets in, 456, 458 pitch-angle in, 457 speed of, 450 useful effect of, 467 weight of water on, 452 Packing, cup-leather, 336 hemp, 336 Parabolic path of jet, 25 Paraboloidal surface, 20 Paris sewer formula, 246 Pastal's press, 336 Path of fluid particle in turbine, 486 Pelton wheel, 486 Pendulum, hydrometric, 308 Permanent regime, I Perrodil's hydrodynamometer, 308 Piezometer, 12 Piobert, 403 Pipe connecting three reservoirs, branched, 191, 200 ; two reser- voirs, 162 equivalent uniform, 186 flow assumptions, 133 flow diagrams, 144 flow in frictionless, 27 Williams' experiments on flow in, 206 of uniform section, flow in, 133 of varying section, 184 thickness of, 158, 159 variation of velocity in transverse section of, 202 Pipe-flow, effect of inclination on, 138 Pitch-back wheel, 472 Pitot tube, 302 Plane layers, motion in, 2 Poiseuille, 128, 131 Poncelet, 48, 418 Poncelet wheel, 424; design of, 433 Position, influence of pipe's, 137 Practical coefficients in centrifugal pumps, 569; turbines, 519 Piess, Bramah's, 336 Baling, 337 hydraulic, 336 Pressure, centre of, xiv due to shock, 160 Pressure-head, n on cylindrical body in pipe, 406 Pressure on thin plate in pipe, 404 of fluids, xiii Prony, 246, 248, 259 Propeller, jet, 373 Pumps, centrifugal, 547; analysis of, 553; vortex-chamber in, 565 Punching bear, 339 Radiating current, 72 Ram, hydraulic, 335 Rayleigh, Lord, 48 Reaction, 373 Reaction wheel, efficiency of, 376 Rectangular orifices, 78 Regime, permanent, i Reservoir sluices, 97 Reservoirs, branched pipe connect- ing three, 191, 200 orifice fed by two, 195 pipe connecting two, 162 Resistance of ships, 131 of motion of solids, 402 Retarding effect of air, etc., in chan- nel flow, 224 Revy's meter, 306 Reynolds, 129, 130, 139, 141 Rhine, experiments on, 247, 262, 266 Ring-nozzle, 61 River-bends, 269 Riveter, portable, 338 Rotation of fluids, 17 Riihlmann, 285, 286, 293 Sagebien wheels, 449 Saone, experiments on, 257 Schiele turbine, 208 Schonheyder's meter, 257, 266 Segner, 375 Seine, experiments on, 257, 266 Sharp-edge orifices, 22 Ships, resistance of, 131 Shock, energy due to, 55 loss of energy in, 55 pressure due to, 160 Simpson's rule, 309 Siphon, 181 inverted, 182 Slotte, 269 Sluice in cylindrical pipe, 169 in rectangular pipe, 169 loss of head due to a, 169 Sluices, 437 reservoir, 97 Smith, Hamilton, Jun., 87 Snow, weight of, 3 Sonnet, 260 Specific gravity, xiii Spiral flow of water, 75 Standing wave, 281 Steady flow in channels of constant section, 221 INDEX. Steady motion, i ; in pipe of uni- form section, 133 Steady varied motion, differential equation of, 202 Stearns, 89 Storage of energy, 340 Stream line, 2 Strickland, 49 St. Venant, 248 Suction-tube, theory of, 529 Surface-floats, 300 Surface-friction in pipes, 126 slope in channels, 226 tension, 49 velocity, 258-265 Tables of backwater function, 290, 291, 292 bottom velocities 269 Castel's results, 69 coefficients of discharge, 39, 40 coefficients of weir discharge by Fteley & Stearns, 89 density of water, 4 discharge through Miner's Inch, 46 discharge through nozzles, 177, 178 elasticity of volume of water, 6 erosion and viscosity, 269 expansion of volume of water, 6 expansion of water, 4 frictional losses in hose, 178 maximum velocities, 269 c and y in v cm x ti, 153 showing best relative dimensions for trapezoidal section, 233 slopes and mean velocities, 227 slopes of. trapezoidal section, 231 values of c and b in Bazin's form- ulae, 311 to 322 values of -^ for centrifugal pumps, 556 values of y in Bazin's formula, 250 TJ 1 r values of for turbines, 503 Vg& values of m and n in Q= m(h-\- n), 316 values of n in Ganguillet & Kut- ter's formula, 251 viscosity of water and mercury, 269 values of c and b in channel form- ulae, Ganguillet & Kutter, 323- 326 Tables, values of c and b in Man- ning's formula, 327 Tachometer, 308 Tadini, 248 Tank, experimental, 29 Tension, surface, 49 Theory of suction or draft tube, 529 of turbines, 497 Thjbault, 403 Thickness of hydraulic pipes and cylinders, 337-344 Thomson, James, 77, 93, 269 Thomson's turbine, 565 Throttle-valve, loss of head due to, 169 Thrupp, 139 Time of emptying and filling a canal-lock, 50 Torricelli's theorem, 24 Transmission of energy by hydrau- lic pressure, 136 Trautwine, 89 Tru ngular notch, 92 Tub-wheel, 387 Turbine, axial-flow, 490, 494 Borda's, 382 Boyden's, 491 centrifugal, 393 combined, 495 efficiency of, 501, 510, 519 Fontaine's, 494 Fourneyron, 491 impulse or Girard, 482, 507, 513, 517 inward-flow, 491, 493 jet, 400 Jonval, 494 limit, 494 losses of effect in, 531 mixed-flow, 490, 495 outward-flow, 491 parallel-flow, 494 practical values of velocities in, 519 radial-flow, 490 reaction, 482, 516 Schiele, 495 Scotch, 375 Segner, 376 Swain's, 495 tangential, 393 theory of, 497 Thomson, 491, 493 useful work of, 501 ventilated 483 vortex, 49r, 493 Whitelaw, 375 INDEX. 583 Tutton, 146, 253, 289, 291, 292 Tweddell's differential accumulator, 342 Undershot-wheel, 416 Undershot wheel, actual delivery in ft.-lbs. of, 423 depth of crown of, 431 efficiency of, 417,420; Poncelet, 428 form of course of, 429 in a straight race, 418 losses of effect with, 421 modifications to increase efficiency of, 423 number of buckets in, 419 Poncelet's, 424; efficiency of, 428 useful work of, 417, 420 with flat vanes, 417 Uniform main, equivalent, 186 Unwin, 403 Useful constants, xvii Vallot, 143 Values of c, x, and y in v = cm x i v , 153 Valve, loss of head due to a, 169 Vane, best form of, 388 cup, 367 Velocity, bottom, 260, 266 critical, 129 curve in a channel, 257 formulae, Bazin's, 266 formulae, Boileau's, 268 maximum, 260, 267 mean, 258, 265 mid-depth, 265 of whirl, 498 rod, 301 surface, 258, 265 variation of, 257 Velocities in turbines, practical values of, 519 Vena contracta, 23 Venant, St., 248 Ventilated buckets, 472 Venturi, water-meter, 16 Vessels in motion, orifice in, 26 Virtual fall, 13 slope, 13 Viscosities, table of, 269 Viscosity, 264 Meyer's formula for, 269 Slotte's formula for, 269 Volute of centrifugal pump, 558 Vortex, circular, 74 compound, 76 free, 74 free-spiral, 75 forced, 75 motion, 74 Water, pressure of, 6 ) weight of, 2 Water-barometer, 7 Water-meter, 207 Water-pressure engine, 347 Water-wheels, classification of ver- tical, 416 Wave propagation, velocity of, 161 Weight of fresh water, 3 of ice, 3 of salt water, 3 Weir, 83 Bazin's flow-over, 99 Beam, 104, 107 broad-crested, 94, 106 drowned, 88, 106 inclined, 89 rectangular, with end contrac- tions, 86 ; without end contrac- tions, 85 sharp-crested, 99, 107 submerged, 88 Weisbach, 36, 60, 166 Weser, experiments on, 247 Wheel, breast, 440 hurdy-gurdy, 485 in straight race, 418 jet reaction, 375 overshot, 450 Pelton, 486 pitch-back, 472 Poncelet, 424 Sagebien, 449 undershot, 416 Whirling fluids, 19 Whirlpool-chamber, 76 Whirl, velocity of, 519 ., Whitelaw, 375 Williams, 206 Woltmann, 247 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. SEP 131934 MAR 18 t937 5941 W MAR 2 8 196859 LD 21-100m-7, , r YC 67953 UNIVERSITY OF CALIFORNIA LIBRARY r* ' \ 1 V