o.. (50) where pa is the greater pressure and pi the less, and the flow is from AO towards AI. By replacing W and H, Hence the initial velocity in the pipe is When I is great, log po/pi is comparatively small, and then «o = V [(sCTm/?ty\(pt?~ pt*)lpis con- stant for all sections, and SI is constant; therefore v must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the bed has a uniform slope. In that case the motion is uniform, the depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section to section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections. In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer 0606 (fig. 102) between sections normal AND CANALS] HYDRAULICS 69 to the direction of motion is treated as sliding down the channel to a'a'b'b' without deformation. The component of the weight parallel to the channel bed balances the friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the stream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean variation of the coefficient of friction with the velocity, proposed an expression of the form r=od+0/t>), (5) and from 255 experiments obtained for the constants the values 0 = 0-007409; # = 0-1920. This gives the following values at different velocities: — v = r= o-3 0-01215 o-5 0-01025 0-7 0-00944 i 0-00883 ll 0-00836 2 O-OO8I2 3 0-90788 5 0-00769 7 0-00761 10 0-00755 15 0-00750 velocity of the stream, and is not in any simple relation with it, for channels of different forms. The theory is therefore obviously based on an imperfect hypothesis. How- ever, by taking variable values for the coefficient of friction, the errors of the ordinary formulae are to a great extent neutralized, and they may be used without leading to practical errors. Formulae have been obtained based on less re- stricted hypotheses, but at present they are not practically so reliable, and are more complicated than the formulae obtained in the manner described above. § 96. Steady Flow of Water with Uniform Velocity in 'Channels of Constant Section. — Let aa', bb' (fig. 103) be two cross sections normal to the direction of motion at a distance dl. Since the mass aa'bb' moves uniformly, the external forces acting on it are in equilibrium. Let & be the area of the cross sections, \ the wetted perimeter, FIG. 102. FIG. 103. pq+qr+rs, of a section. Then the quantity m = tt/x is termed the hydraulic mean depth of the section. Let v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the slope or fall of the stream in feet, per foot, being the ratio bc/ab. The external forces acting on aa'bb' parallel to the direction of motion are three: — (a) The pressures on aa' and bb', which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same. (6) The component of the weight W of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aa'bb' is Gildl, and the com- ponent of the weight in the direction of motion is GSldl X the cosine of the angle between Wg and ab, that is, GQdl cos abc = Gttdl bc/ab = GOidl. (c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area \dl of rubbing surface and to a function of the velocity which may be written _f(i>) ; /(») being the friction per sq. ft. at a velocity v. Hence the friction is — \dlf(ji). Equating the sum of the forces to zero, (i) But it has been already shown (§ 66) that/(ti) = .'. fi>2/2g = mt. ' (2) This may be put in the form " = V(2g/f)V("»)=cV(»«); (20) where c is a coefficient depending on the roughness and form of the channel. The coefficient of friction f varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the error of neglecting the variations of velocity in the cross section. A common mean value assumed for f is 0-00757. The range of values will be discussed presently. It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. If/ is the fall in feet per mile> / = 52801. Putting this and the above value of f in (20), we get the very simple and long-known approximate formula for the mean velocity of a stream — » = HV(2m/). (3) The flow down the stream per second, or discharge of the stream, Q = 0» = n»V (mi)- (4) § 97. Coefficient of Friction for Open Channels. — Various ex- pressions have been proposed for the coefficient of friction for ' annels as for pipes. Weisbach, giving attention chiefly to the In using this value of f when v is not known, it is best to proceed by approximation. § 98. Darcy and Bazin's Expression for the Coefficient of Friction, — Darcy and Bazin's researches have shown that f varies very greatly for different degrees of roughness of the channel bed, and that it also varies with the dimensions of the channel. They give for f an empirical expression (similar to that for pipes) of the form f = a(i-hS/m); (6) where m is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction : — Kind of Channel. I. Very smooth channels, sides of smooth cement or planed timber .... II. Smooth channels, sides of ashlar, brick- work, planks ... ... III. Rough channels, sides of rubble masonry or pitched with stone IV. Very rough canals in earth .... V. Torrential streams encumbered with detritus 0-00294 0-00373 0-00471 0-00549 0-00785 o-io 0-23 0-82 4-10 5-74 .The last values (Class V.) are not Darcy and Bazin's, but are taken from experiments by Ganguillet and Kutter on Swiss streams. The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy and Bazin's value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (20) above, D = cV (mi), where c = V (2g/f), the following table gives values of c for channels of different degrees of roughness, and for such values of the hydraulic mean depths as are likely to occur in practical calculations : — Values ofc in v = cV (mi), deduced from Darcy and Bazin's Values. . JS J4 .a 5 e •T? O £ x 4> . eo A 3 • (3 it's •A r»» oj • C -fl ^ a 3 Hydraulic Me; Depth = m. Very Smooth Channels. Cem< Smooth Chann( Ashlar or Brickw Rough Channe Rubble Masonr Very Rough Chan 4 Canals in Eart Excessively Roi Channels encui bered with Detri Hydraulic Mei Depth = m. Very Smoott Channels. Ceme Smooth Channe Ashlar or Brickw ll Is £3 Very Rough Chan Canals in Eart Excessively Rou Channels encui bered with Detri •25 125 95 57 26 18-5 8-5 H7 130 112 89 •5 135 no 72 36 25-6 9-0 H7 130 112 90 71 •75 139 116 81 42 30-8 9'5 147 130 112 90 I-O 141 119 •87 48 34'9 IO-O 147 130 112 91 72 1-5 143 122 94 56 41-2 II 147 130 113 92 2-O 144 124 98 62 46-0 12 147 130 113 93 74 2-5 H5 126 IOI 67 13 147 130 113 94 3-0 145 126 104 70 53 14 H7 130 "3 95 3-5 146 127 105 73 15 H7 130 114 96 77 4-0 146 128 106 76 58 16 147 130 114 97 4'5 146 128 107 78 17 H7 130 114 97 146 128 1 08 80 62 18 147 130 114 98 5'5 146 129 109 82 20 H7 131 114 98 80 6-0 147 129 IIO 84 65 25 148 131 US IOO 6-5 129 IIO 85 30 148 131 US 102 83 7-0 147 129 IIO 86 67 4° 148 131 116 103 85 7-5 147 129 III 87 50 I48 116 IO4 86 8-0 147 130 III 88 69 00 148 131 117 1 08 9i § 99. Ganguillet and Kutter's Modified Darcy Formula. — Starting from the general expression v — c^mi, Ganguillet and Kutter examined the variations of c for a wider variety of cases than those discussed by Darcy and Bazin. Darcy and Bazin's experiments were confined to channels of moderate section, and to a limited variation of slope. Ganguillet and Kutter brought into the dis- cussion two very distinct and important additional series of results. The gaugings of the Mississippi by A. A. Humphreys and L. H. Abbot afford data of discharge for the case of a stream of exception- ally large section and of very low slope. On the other hand, their own measurements of the flow in the regulated channels of some 7o HYDRAULICS [FLOW IN RIVERS Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy and Bazin's experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its rough- ness of surface. Plotting values of c for channels of different in- clination appeared to indicate that it also depended on the slope of the stream. Taking the Mississippi data only, they found £ = 256 for an inclination of 0-0034 per thousand, = 154 „ „ 0-02 so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge. In small rivers, on the other hand, the values of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of rough- ness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy and Bazin are different for each of the classes of channels of different roughness, even when the dimensions of the channel are infinite. But, as it is much more probable that the influence of the nature of the sides diminishes indefinitely as the channel is larger, this must be regarded as a defect in their formula. Comparing their own measurements in torrential streams in Switzerland with those of Darcy and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus. To obtain a new expression for the coefficient in the formula » = V (2g/f ) V (mi) = c V (mi) , Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form could be made to fit the experiments somewhat better than Darcy's expression. Inverting this, we get I/c=I/a+/3/oVm, an equation to a straight line having i/^m for abscissa, i/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is /3/o. Plotting the experimental values of l/c and i/V»», the points so found indicated a curved rather than a straight line, so that 0 must depend on a. After much comparison the following form was arrived at — where n is a coefficient depending only on the roughness of the sides of the channel, and A and / are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases. Let A. = a+p/i. Then c = (a+l/n+p/i)/(l+(a+p/i)nHm}, the form of the expression for c ultimately adopted by Ganguillet and Kutter. For the constants a, I, p Ganguillet and Kutter obtain the values 23, I and 0-00155 f°r metrical measures, or 41-6, 1-811 and 0-00281 for English feet. The coefficient of roughness n is found to vary from 0-008 to 0-050 for either metrical or English measures. The most practically useful values of the coefficient of roughness n are given in the following table : — Nature of Sides of Channel. Coefficient of Roughness n. Well-planed timber ......... 0-009 Cement plaster .......... o-oio Plaster of cement with one-third sand .... o-oi I Unplaned planks .......... 0-012 Ashlar ana brickwork ......... 0-013 Canvas on frames ......... 0-015 Rubble masonry .......... 0-017 Canals in very firm gravel ....... 0-020 Rivers and canals in perfect order, free from stones ) or weeds ........... I 0-025 Rivers and canals in moderately good order, not } quite free from stones and weeds . . . . { o'°3° Rivers and canals in bad order, with weeds and / detritus ............ \ °'°35 Torrential streams encumbered with detritus . . 0-050 Ganguillet and Kutter's formula is so cumbrous that it is difficult to use without the aid of tables. Lewis D'A. Jackson published complete and extensive tables for facilitating the use of the Ganguillet and Kutter formula (Canal Values of M fo r n = O-OIO O-OI2 0-015 0-017 O-O2O 0-025 0-030 •OOOOI 3-2260 3-8712 4-8390 5-4842 6-4520 8-0650 9-6780 •OOOO2 1-8210 2-1852 2-73I5 3-0957 3-6420 4-5525 5-4630 •OOOO4 1-1185 1-3422 1-6777 1-9014 2-2370 2-7962 3-3555 •OOOO6 0-8843 1-0612 1-3264 I-5033 1-7686 2-2107 2-6529 •00008 0-7672 0-9206 1-1508 1-3042 1-5344 •9180 2-3016 •OOOIO 0-6970 0-8364 1-0455 1-1849 1-3940 •7425 2-0910 •00025 0-5284 0-6341 0-7926 0-8983 1-0568 •3210 •5852 •00050 0-4722 0-5666 0-7083 0-8027 0-9444 •1805 •4166 •00075 0-4535 0-5442 0-6802 0-7709 0-9070 •1337 •3605 •OOIOO 0-4441 0-5329 0-6661 0-7550 0-8882 •IIO2 •3323 •OO2OO 0-4300 0-5160 0-6450 0-7310 0-8600 •0750 •2900 •00300 0-4254 0-5105 0-6381 0-7232 0-8508 •0635 •2762 and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form : — M = M(4i-6+o-oo28i/»); » = (Vw/w)l(M + i-8ii)/(M+Vw))V (mi). The following table gives a selection of values of M, taken from Jackson's tables: — • A difficulty in the use of this formula is the selection of the co- efficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from o 0163 to 0-0301. For natural channels or rivers n varies from 0-020 to 0-035. In Jackson's opinion even Kutter's numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification : — I. Canals in very firm gravel, in perfect order n=o-O2 II. Canals in earth, above the average in order n=o-O225 III. Canals in earth, in fair order .... n = 0-025 IV. Canals in earth, below the average in order n = 0-0275 V. Canals in earth, in rather bad order, partially } overgrown with weeds and obstructed by >n = 0-03 detritus . . ...... ' Ganguillet and Kutter's formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it i» an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is com- plicated it must be more accurate than simpler formulae. Com- parison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results. § too. Bazin's New Formula. — Bazin subsequently re-examined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (£tude d'une nouvelle formule, Paris, 1898). He points out that Darcy's original formula, which is of the form mi/i? = a+fi/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/v* tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that V (mi/ti3) = o + /3/V»w, so that if o is a constant mifv" tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Kutter's. Putting the equation in the form ft>2/2g = mi, f = 0-002594(1 +7/V>w), where y has the following values: — I. Very smooth sides, cement, planed plank, 7 = 0-109 II. Smooth sides, planks, brickwork .... 0-290 III. Rubble masonry sides ....... 0-833 IV. Sides of very smooth earth, or pitching . . 1-539 V. Canals in earth in ordinary condition . . . 2-353 VI. Ca'nals in earth exceptionally rough . . . 3-168, § 101. The Vertical Velocity Curve. — If at each point along a vertical representing the depth of a stream, the velocity at that point is plotted horizontally, the curve obtained is the vertical velocity curve and it has been shown by many observations that it approximates to a parabola with horizontal axis. The vertex of the parabola is at the level of the greatest velocity. Thus in fig. 104 OA is the vertical at which velocities are observed; va is the sur- face; v, the maximum and Vd the bottom velocity. B C D is the vertical velocity curve which corresponds with a parabola having its vertex at C. The mean velocity at the vertical is The Horizontal -Velocity Curve. — Similarly if at each point along a horizontal representing the width of the stream the velocities are AND CANALS] HYDRAULICS plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In un- symmetrical sections the greatest velocity is at the point where the stream is deepest, and the general form of the horizontal velocity curve is roughly similar to the section of the stream. § 1 02. Curves or Contours of Equal Velocity. — If velocities are observed at a number of points at different widths and depths in a stream, it is possible to draw curves on the cross section through points at which the velocity is the same. These repre- sent contours of a solid, the volume of which is the discharge of the stream per second. Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin's gaugings. § 103. Experimental Observations on the Vertical Velocity Curve. — A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that FIG. 104. «f g b ^- Contours of Equal Velocity FIG. 105. though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded. In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth d, at the centre of the stream has been found to vary from o to 0-3^. § 104. Influence of the Wind. — In the experiments on the Missis- sippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at $,ths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the para- bola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained. It is not difficult to understand that a wind acting on surface ripples or waves should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experi- ments, P. P. Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a re- tarding action. A much more probable explanation of the diminution of the velocity at and near the free surface is that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there. Influence of the Wind on the Depth at which the Maximum Velocity is found. — In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from o for a calm to 10 for a hurricane. Arranging the velocity curves in three sets — (i) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream — it was found that an up-stream wind lowered, and a down-stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at -ftths of the total depth from the surface for all conditions of the stream. Let h' be the depth of the axis of the parabola, m the hydraulic mean depth, / the number expressing the force of the wind, which may range from + io to — 10, positive if the wind is up stream, negative if it is down stream. Then Humphreys and Abbot find their results agree with the expression h'/m =0-317 ±O-O6/. Fig. 106 shows the parabolic velocity curves according to the American observers for calm weather, and for an up- or down-stream wind of a force represented by 4. . FIG. 106. It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0-85 to 0-92 of the surface velocity at that vertical, and on the average if v, is the surface and »„, the mean velocity at a vertical vm = %vc, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at 0-66 of the depth ; G. H. L. Hagen and H. Heinemann at 0-56 to 0-58 of the depth. The mean of observations by various observers gave the mean velocity at from 0-587 to 0-62 of the depth, the average of all being almost exactly 0-6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If vmd is the mid-depth velocity, then on the average vm=o-<)8vmd. § 105. Mean Velocity on a Vertical from Two Velocity Observations. — A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let v, be the surface, vm the mean, and Vzd the velocity at the depth xd ; then § 1 06. Ratio of Mean to Greatest Surface Velocity, for the whole Cross Section in Trapezoidal Channels^. — It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regular section the mean velocity for the whole section varies from 0-7 to 0-85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let va be the greatest surface velocity, vm the mean velocity of the stream. Then, according to Bazin, »«•=»„— 25-4V (»»')• But vm = c^l(mi), where c is a coefficient, the values of which have been already given in the table in § 98. Hence HYDRAULICS [FLOW IN RIVERS Values of Coefficient c/(c+25~4) in the Formula vm = c Hydraulic Mean Depth =m. Very Smooth Channels. Cement. Smooth Channels. Ashlar or Brickwork. Rough Channels. Rubble Masonry. Very Rough Channels. Canals in Earth. Channels encumbered with Detritus. 0-25 •83 •79 •69 •51 •42 °-5 •84 •81 •74 •58 •50 o-75 •84 •82 •76 •63 •55 I-O •85 •77 •65 •58 2-O •83 •79 •71 •64 3-o •80 •73 •67 4-0 •8: •75 •70 5-o •76 •7i 6-0 •84 •77 •72 7-0 •78 •73 8-0 9-0 •82 •74 IO-O 15-0 •79 •75 2O-O •80 •76 30-0 •82 •77 4O-O 50-0 oo •79 § 107. River Bends. — In rivers flowing in alluvial plains, the wind- ings which already exist tend to increase in curvature by the scouring away of material from the outer bank and the deposition of detritus along the .inner bank. The sinuosities sometimes increase till a loop is formed with only a narrow strip of land between the two encroaching branches of the river. Finally a " cut off " may occur, a waterway being opened through the strip of land and the loop left separated from the stream, forming a horse- shoe shaped lagoon or marsh. Professor James Thomson pointed out (Proc. Roy. Soc., 1877, P- 356; Proc. Inst. of Mech. Eng., 1879, p. 456) that the usual supposi- tion is that the water jy tending to go forwards in a straight line rushes against the outer bank and scours it, at the same time creating de- posits at the inner bank. That view is very far from a complete account of the matter, and Pro- fessor Thomson gave a p much more ingenious 10. 107. account of the action at the bend, which he completely confirmed by experiment. When water moves round a circular curve under the action of gravity only, it takes a motion like that in a free vortex. Its velocity is greater parallel to the axis of the stream at the inner than at the outer side of the bend. Hence the scouring at the outer side and the deposit at the inner side of the bend are not due to mere difference of velocity of flow in the general direction of the stream; but, in virtue of the centrifugal force, the water passing round the bend presses outwards, and the free surface in a radial cross section has a slope from the inner side upwards to the outer side (fig. 108). For the greater part of the water flowing in curved paths, this difference of pressure produces no tendency to transverse motion. But the water im- InncrBanlt Outer Bank Section at M N. FlG. 108. mediately in contact with the rough bot- tom and sides of the channel is retarded, and its centrifugal force is insufficient to balance the pressure due to the greater depth at the outside of the bend. It there- fore flows inwards towards the innet side of the bend, carrying with it detritus which is deposited at the inner bank. Con- jointly with this flow inwards along the bottom and sides, the general mass of water must flow outwards to take its place. Fig. 107 shows the directions of flow as observed in a small artificial stream, by means of light seeds and specks of aniline dye. The lines CC show the directions of flow immediately in contact with the sides and bottom. The dotted line AB shows the direction of motion of floating particles on the surface of the stream. § 1 08. Discharge of a River when flowing at different Depths. — When frequent observations must be made on the flow of a river or canal, the depth of which varies at different times, it is very convenient to have to observe the depth only. A formula can be established giving the flow in terms of the depth. Let Q be the discharge in cubic feet per second ; H the depth of the river in some straight and uniform part. Then Q = oH+6H2, where the constants a and b must be found by preliminary gaugings in different con- ditions of the river. M. C. Moquerey found for part of the upper Sa&ne, Q=64'7H+8-2H2 in metric measures, or Q = 696H+26-8H« in English measures. § 109. Forms of Section of Channels. — The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 109), a form now often adopted from the facility with which it can be FlG. 109. executed in concrete. It has the advantage that the rubbing surface is less in proportion to the area than in any other form. Wooden channels or flumes, of which there are examples on a large scale in America, are rectangular in section, and the same form is adopted for wrought and cast-iron aqueducts. Channels built with brickwork or masonry may be also rectangular, but they are often trapezoidal, and are always so if the sides are pitched with masonry laid dry. In a trapezoidal channel, let b (fig. no) FIG. no. be the bottom breadth, 60 the top breadth, d the depth, and let the slope of the sides be n horizontal to I vertical. Then the area of section is U = (b+nd)d = (ba — nd)d, and the wetted perimeter When a channel is simply excavated in earth it is always originally trapezoidal, though it becomes more or less rounded in course of time. 'The slope of the sides then depends on the stability of the earth, a slope of 2 to I being the one most commonly adopted. Figs, in, 112 show the form of canals excavated in earth, the former being the section of a navigation canal and the latter the section of an irrigation canal. § 1 10. Channels of Circular Section. — The following short table facilitates calculations of the discharge with different depths of water in the channel. Let r be the radius of the channel section; then for a depth of water = xr, the hydraulic mean radius is itr and the area of section of the waterway w2, where K, MI and v have the following values: — terms of radius . . J " •OS .10 •15 .20 •25 •30 •35 .40 •45 •50 •55 .60 • 65 •70 •75 .80 .85 .90 • 95 I.O Hydraulic mean depth / in terms of radius . ) .00668 .0331 •0523 .0963 .1278 •'574 .1852 .2142 .242 .269 •253 •320 •343 .365 .387 .408 .429 •449 .466 .484 .500 Waterway in terms oO _ square of radius . . ) .00180 .0211 .0508 .1067 .1651 .228 .204 •370 •450 •532 .614 • 700 •705 • 885 •079 1-075 I.I75 1.276 1371 1.470 I.57I AND CANALS] HYDRAULICS 73 § III. Egg-Shaped Channels or Sewers. — In sewers for discharging storm water and house drainage the volume of flow is extremely variable; and there is a great liability for deposits to be left when the flow is small, which are not removed during the short periods when the flow is large. The sewer in consequence becomes choked. In Bank could be found satisfying the foregoing conditions. To render the problem determinate, let it be remembered that, since for a given discharge to -J x< other things being the same, the amount of excavation will be least for that channel which has the least wetted perimeter. Let d be the depth and b the bottom width of the channel, and let the sides slope n horizontal to I vertical (fig. 114), then In Cattincf Both J2 and x are to be minima. Differentiating, and equating to zero. (db/dd+n)d+b+nd = o, FIG. in. — Scale 20 ft. = i in. eliminating dbjdd, But Inserting the value of b, j» 120-O—r.- FIG. 112. — Scale 80 ft. = i in. To obtain uniform scouring action, the velocity of flow should be constant or nearly so; a complete uniformity of velocity cannot be obtained with any form of section suitable for sewers, but an ap- proximation to uniform velocity is obtained by making the sewers of oval section. Various forms of oval have been suggested, the simplest being one in which the radius of the crown is double the radius of the invert, and the greatest width is two- thirds the height. The section of such a sewer is shown in fig. 113, the numbers marked on the figure being proportional •-(. i\ ./ •»' numbers. § 112. Problems on Channels in which the Flow is Steady and at Uniform Velocity. — The general equations given in §§ 96, 98 are J- = a(l+0/m); (l) fi>2/2g = mi ; (2 FIG. 113. Q=to. (3 Problem I. — Given the transverse section of stream and dis- charge, to find the slope. From the dimensions of the section find tt and m; from (i) find f, from (3) find », and lastly from (2) find i. Problem II. — Given the transverse section and slope, to find the discharge. Find r from (2), then Q from (3). Problem III. — Given the discharge and slope, and either the breadth, depth, or general form of the section of the channel, to determine its remaining dimensions. This must generally be solved by approximations. A breadth or depth or both are chosen, and the discharge calculated. If this is greater than the given discharge, the dimensions are reduced and the discharge recalculated. Since m lies generally between the limits m = d and m = %d, where d is the depth of the stream, and since, moreover, the velocity varies as V (m) so that an error in the value of m leads only to a much less error in the value of the velocity calculated from it, we may proceed thus. Assume a value for m, and calculate v from it. Let iii be this first approximation to v. Then Qjvi is a first approxi- mation to 12, say Qi. With this value of ft design the section of the annel ; calculate a second value for m ; calculate from it a second value of v, and from that a second value for fi. Repeat the process till the succes- sive values of m approxi- mately coincide. § 113. Problem IV. Most Economical Form of Channel p for given Side Slopes. — Sup- pose the channel is to be trapezoidal in section (fig. 114), and that the sides are to have a given slope. Let the longitudinal slope of the stream be given, and also the mean velocity. An infinite number of channels That is, with given side slopes, the section is least for a given discharge when the hydraulic mean depth is half the actual depth. A simple construction gives the form of the channel which fulfils this condition, for it can be shown that when m = \d, the sides of the channel are tangential to a semicircle drawn on the water line. Since £)/x = \d, therefore Q = Jx^- (i) Let ABCD be the channel (fig. 115); from E'the'centre of AD drop perpendiculars EF, EG, EH on the sides. AB=CD=a; BC=6; EF = EH=c; and EG=d. H = area AEB + BEC+CED, = ac -\- \ba. Putting these values in (i), = (a + \V)d ; and hence c = d. E B G C FIG. 115. That is, EF, EG, EH are all equal, hence a semicircle struck from E with radius equal to the depth of the stream will pass through F and H and be tangential to the sides of / the channel. To draw the channel, describe a semicircle on a horizontal line with radius = depth of channel. i* & x The bottom will be a FIG. 116. horizontal tangent of that semicircle, and the sides tangents drawn at the required side slopes. The above result may be obtained thus (fig. 1 16) : — (i) (2) (3) From (i) and (2), This will be a minimum for dx/dd =fi/and the mean velocity. For the whole cross section, The mass of fluid passing through the element of section u, in 8 seconds, is (G/g)un0, and its kinetic energy is (G/2g)u®38. For the whole section, the kinetic energy of the mass AoBoCoDo passing in 8 seconds is (G9/2g)So«i3 = The factor 3Ut>+w is equal to 2«o-ff, a quantity necessarily positive. Consequently 2ufs> QnUtf, and consequently the kinetic energy of AoBoCoDo is greater than which would be its value if all the particles passing the section had the same velocity «o. Let the kinetic energy be taken at a(Ge/2g)a«,,>» = a(G0/2g)Q«o2, where o is a corrective factor, the value of which was estimated by J. B. C. J. B61anger at i-i.1 Its precise value is not of great im- portance. In a similar way we should obtain for the kinetic energy of AiBiCiD: the expression o(G0/2g)n,«i> = a(G0/2g)Q«i8, where ft, Ui are the section and mean velocity at AiBi, and where a may be taken to have the same value as before without any im- portant error. Hence the change of kinetic energy in the whole mass AoBoAiBi in 8 seconds is a(G»/2g)Q(w,«-itf). (i) Motive Work of the Weight and Pressures.— Consider a small filament OoOi which comes in 6 seconds to CoCi. The work done by gravity during that movement is the same as if the portion aoCt, were carried to aid. Let dQff be the volume of OoCo or a\c\, and ya, y\ the depths of ac, ot from the surface of the stream. Then the volume 1 Boussinesq has shown that this mode of determining the corrective factor a is not satisfactory. AND CANALS] HYDRAULICS 75 dQ6 or GdQO pounds falls through a vertical height 2+3-1— 3*0, and the work done by gravity is GdQ6(z +311-3-0). Putting pa for atmospheric pressure, the whole pressure per unit of area at oo is Gyo+pa, and that at ai is — (Gyi+p,). The work of these pressures is G(3-o+£a/G -3-! -pt/QdQe = G(3-o -yi)dQ6. Adding this to the work of gravity, the whole work is GzdQO; or, for the whole cross section, GzQO. (2) Work expended in Overcoming the Friction of the Stream Bed. — Let A'B', A"B" be two cross sections at distances s and s+ds from AoBo. Between these sections the velocity may be treated as uni- form, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A'B' and A"B"is where it, x, ® are the mean velocity, wetted perimeter, and section at A'B'. Hence the whole work lost in friction from AoBo to AiBi will be (3) Equating the work given in (2) and (3) to the change of kinetic energy given in (i), a(GQ0/2g) (M,2 -Mo2) = GQaS -GQ0/«'f(« .'. Z = a(«i' -Wo2)/2g+ § 116. Fundamental Differential Equation of SteadyVariedMotion. — Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, Oi&j, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of FIG. 120. a and 01, may be written dz. Also, if we write u for uo, and u+du for MI, the term (Mo2 — «i!)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is

, in which case u is small, the numerator becomes equal to i. For a value H of h given by the equation H=fM2/2gi, we fall upon the case of uniform motion. The results just stated may be tabulated thus : — For h = o, H, >H, oo, the numerator has the value — °o, o, > o, i. Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit — oo . As h becomes very large and u consequently very small, the de- nominator tends to the limit i. For h = u?/g, or w = V(gA), the denominator becomes zero. Hence, tabulating these results as before : — For h = o, u*/g, >M2/g, oo, the denominator becomes — oo, o, > o, i. § 118. Case i. — Suppose h>u?/g, and also ft>H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let BoBi be the bed, CcCi a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface . 122. AoAi of the stream is above CoCi, and it has just been shown that the depth of the stream increases from Bo towards BI. But going up stream h approaches more and more nearly the value H, and there- fore dh/ds approaches the limit o, or the surface of the stream is asymptotic to CoCi. Going down stream h increases and u diminishes, thenumeratorand denominator of thefraction(i — f«2/2gt7i)/(l — M2/gA) both tend towards the limit i, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line DoDi. The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a weir. The raising of the water surface above the level CoCi is termed the backwater due to the weir. § 119. Case 2. — Suppose A>«2/g, and also A u2/g diminishes; the denominator of the frac- tion (i—fu2/2gih)l(i-u''/gh) tends to the limit zero, and con- sequently dh/ds tends to » . That is, down stream AoAi tends to a direction perpendicular to the bed. Before, however, this limit was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion, which would make the influence of internal friction in the fluid too important to be neglected. A stream surface of this form may be pro- duced if there is an abrupt fall in the bed of the stream (fig. 124). On the Ganges canal, as orig- inally con- structed, there were abrupt falls precisely FlG. 124. of this kind, and it appears that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some miles up stream, was not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and considerable damage was done to the works. " When the canal was first opened the water was allowed to pass freely over the crests of the overfalls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles up rapidly followed, and it soon became apparent that means must be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were accord- ingly fixed in the grooves above the bridge arches, or temporary weirs were formed over which the water was allowed to fall ; in some cases the surface of the water was thus raised above its normal height, causing a backwater in the channel above " (Crofton's Report on the Ganges Canal, p. 14). Fig. 125 represents in an ex- aggerated form what probably occurred, the diagram being intended FIG. 125. to represent some miles' length of the canal bed above the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water surface CC corresponding to Case I would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA. 8 1 20. Case 3. — Suppose a stream flowing uniformly with a depth « f/2. If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes Ai>«2/g, then for a portion of the stream the depth h will satisfy the con- ditions h H, which are not the same as those assumed in the two previous cases. At some point of the stream above the weir the depth h becomes equal to u2/g, and at that point dh/ds becomes infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surface of the stream, or a standing wave is formed, the conditions of motion in which will be examined presently. It appears that the condition necessary to give rise to a standing wave is that i>?/2. Now f depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin calculated the values of f for channels of different degrees of rough- ness and different depths given in the following table, and the corre- sponding minimum values of i for which the exceptional case of the production of a standing wave may occur. STANDING WAVES § 121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 0-023 ft- P61" loot, he admitted water till it flowed uniformly with a depth of 0-2 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0-95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from 0-2 ft. to 0-56 ft. _ The velocity of the stream in the part un- affected by the obstruction was 5-54 ft. per second. Above the point where the abrupt change of depth occurred u* = 5-54' = 30-7, and gh = 32-2X0-2 =6-44; hence tt2 was>g/t. Just below the abrupt change of depth « = 5-54X0-2/0-56 = 1-97; «2 = 3-88; and gh = 32-2X0-56 = 18-03; hence at this point u', Qi the areas of the cross sections. The force causing change of momentum in the mass abed estimated horizont- ally is simply the difference of the pressures on ab and cd. Putting ho, hi for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(&oJi> — AiQ,) pounds, and the_ impulse in / seconds is G (Wi> — AiJJi) t second pounds. The horizontal change of momentum is the difference of the momenta of cdc'd' and aba'b' ; that is, ON STREAMS AND RIVERS] Hence, equating impulse and change of momentum, HYDRAULICS 77 (i) For simplicity let the section be rectangular, of breadth B and depths Ho and Hi, at the two cross sections considered; then Ao = iHo, and fci = iHi. Hence But, since = Qi«i, we have Hi-Ho). (2) This equation is satisfied if Ho = Hi, which corresponds to the case of uniform motion. Dividing by H0 — HL, the equation becomes (H,/H0)(Ho+H1)=2«o2/g; (3) In Bidone's experiment Mo = 5'54, a"d H=o-2. Hence Hi=o-52, which agrees very well with the observed height. § 122. A standing wave is frequently produced at the foot of a weir. Thus in the ogee falls originally constructed on the Ganges canal a standing wave was observed as shown in fig. 127. The water falling over the weir crest A acquired a very high velocity on the FIG. 127. steep slope AB, and the section of the stream at B became very small. It easily happened, therefore, that at B the depth h -.-00 « o "> <" S S fe£ 8£ gjg &£ 5 °> 29-« 1-08 4-80 6-667-30 9-2+ 9-SO 11-82 12-30 !*•* H-80K-92 17-30 19-SJ 19-80 22-15 22-3O 24-8O 27-30 & S Discharge per Second = Q= 14-10 87 cub'm Carves of equal velocity. Transformation ra/io 10:1 • " 3 t i FIG. 151. sections normal to the plane of fig. 149 given by the diagrams in fig. 150. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let Qo be the area of the cross section of the river, HL BI . . . the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at I ft. per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be ix(no-rA), %x($li+Qi), and so on. Consequently the discharge is The areas J2o, fli . . . are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly *, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be estimated separately, and taken to be the area of its base (the area fin) multiplied by 3 to | its height. Fig. 151 shows the results of one of Harlacher's gaugings worked and pump. It may be noted that constructively pumps are essentially reversed motors. The reciprocating pump is a re- versed pressure engine, and the centrifugal pump a reversed turbine. Hydraulic machine tools are in principle motors com- bined with tools, and they now form an important special class. Water under pressure conveyed in pipes is a convenient and economical means of transmitting energy and distributing it to many scattered working points. Hence large and important hydraulic systems are adopted in which at a central station water is pumped at high pressure into distributing mains, which convey it to various points where it actuates hydraulic motors operating cranes, lifts, dock gates, and in some cases riveting and shearing machines. In this case the head driving the hydraulic machinery is artificially created, and it is the con- venience of distributing power in an easily applied form to distant points which makes the system advantageous. As there is some unavoidable loss in creating an artificial head this system is most suitable for driving machines which work intermittently 86 HYDRAULICS [IMPACT AND REACTION (see POWER TRANSMISSION). The development of electrical methods of transmitting and distributing energy has led to the utilization of many natural waterfalls so situated as to be useless without such a means of transferring the power to points where it can be conveniently applied. In some cases, as at Niagara, the hydraulic power can only be economically developed in very large units, and it can be most conveniently subdivided and distributed by transformation into electrical energy. Partly from the development of new industries such as paper-making from wood pulp and electro-metallurgical processes, which require large amounts of cheap power, partly from the facility with which energy can now be transmitted to great distances electrically, there has been a great increase in the utilization of water-power in countries having natural waterfalls. According to the twelfth census of the United States the total amount of water-power reported as used in manufacturing establishments in that country was 1,130,431 h.p. in 1870; 1,263,343 h.p. in 1890; and 1,727,258 h.p. in 1900. The increase was 8-4% in the decade 1870-1880, 3-1% in 1880-1890, and no less than 36-7% in 1890-1900. The increase is the more striking because in this census the large amounts of hydraulic power which are transmitted electrically are not included. XII. IMPACT AND REACTION OF WATER § 153. When a stream of fluid in steady motion impinges on a solid surface, it presses on the surface with a force equal and opposite to that by which the velocity and direction of motion of the fluid are changed. Generally, in problems on the impact of fluids, it is necessary to neglect the effect of friction between the fluid and the surface on which it moves. During Impact the Velocity of the Fluid relatively to the Surface on which it impinges remains unchanged in Magnitude. — Consider a mass of fluid flowing in contact with a solid surface also in motion, the motion of both fluid and solid being estimated relatively to the earth. Then the motion of the fluid may be resolved into two parts, one a motion equal to that of the solid, and in the same direction, the other a motion relatively to the solid. The motion which the fluid has in common with the solid cannot at all be influenced by the con- tact. The relative component of the motion of the fluid can only be altered in direction, but not in magnitude. The fluid moving in contact with the surface can only have a relative motion parallel to the surface, while the pressure between the fluid and solid, if friction is neglected, is normal to the surface. The pressure therefore can only deviate the fluid, without altering the magnitude of the relative velocity. The unchanged common component and, combined with it, the deviated relative component give the resultant final velocity, which may differ greatly in magnitude and direction from the initial velocity. From the principle of momentum, the impulse of any mass of fluid reaching the surface in any given time is equal to the change of momentum estimated in the same direction. The pressure between the fluid and surface, in^any direction, Is equal to the change of momentum in that direction of so much fluid as reaches the surface in one second. If Po is the pressure in any direction, m the mass of fluid impinging per second, va the change of velocity in the direction of Pa due to impact, then P0=»mi<.. If DI (fig. 152) is the velocity and direction of motion before impact, vi that after impact, then v is the total change of motion due to impact. The resultant pressure of the fluid on the surface is in the direction of v, and is equal to v multiplied by the mass impinging per second. That is, putting P for the resultant pressure, P = mv. Let P be resolved into two components, N and T, normal and tangential to the direction of motion of the solid on which the fluid impinges. Then N is a lateral force producing a pressure on the supports of the solid, T is an effort which does work on the solid. If u is the velocity of the solid, Tit is the work done per second by the fluid in moving the solid surface. Let Q be the volume, and GQ the weight of the fluid impinging per second, and let 1/1 be the initial velocity of the fluid before striking the surface. Then GQvSfeg is the original kinetic energy of Q cub. ft. of fluid, and the efficiency of the stream considered as an arrange- ment for moving the solid surface is § 154. Jet deviated entirely in one Direction. — Geometrical Solution (fig- IS3)- — Suppose a jet of water impinges on a surface ac with a velocity ab, and let it be wholly deviated in planes parallel to the figure. Also let ae be the velocity and direction of motion of the surface. Join eb; then the water moves with respect to the surface in the direction and with the velocity eb. As this relative velocity is unaltered by contact with the surface, take cd = eb, tangent to the surface at c, then cd is the relative motion of the water with respect to the surface at c. Take df equal and parallel to ae. Then/c (obtained by compounding the relative motion of water to surface and common velocity of water and surface) is the absolute velocity and direction FIG. 153. of the water leaving the surface. Take ag equal and parallel to fc. Then, since ab is the initial and ag the final velocity and direction of motion, go is the total change of motion of the water. The resultant pressure on the plane is in the direction gb. Join eg. In the triangle gae, ae is equal and parallel to df, and ag to/c. Hence eg is equal and parallel to cd. But cd — eb = relative motion of water and surface. Hence the change of motion of the water is represented in magnitude and direction by the third side of an isosceles triangle, of which the other sides are equal to the relative velocity of the water and surface, and parallel to the initial and final directions of relative motion. SPECIAL CASES § 155- (l) A Jet impinges on a plane surface at rest, in a direction normal to the plane (fig. 154). — Let a jet whose section is u> impinge with a velocity v on a plane surface at rest, in a direction normal to the plane. The particles approach the plane, are gradually deviated, and finally flow away parallel to the plane, having then no velocity in the original direction of the jet. The quantity of water impinging per second is uv . The pressure on the plane, which is equal to V the change of momentum per second, is (2) // the plane is moving in the direction of the jet with the velocity *=u, the quantity impinging per second is tafyfU). The momentum of this quantity before impact is (G/g)u(v=f=tt)ti. After impact, the water still possesses the velocity =*=« in the direction of the jet; and the momentum, in that direction, of so much water as impinges in one second, after impact, is ±(G/g)u>(»=i=tt)tt. The pressure on the plane, which is the change of momentum per second, is the difference of these quantities or P = (G/g)o>(ti=?=«)2. This differs from the expression obtained in the previous case, in that the relative velocity of the water and plane v*=u is sub- stituted for t». The expression maybe written P = 2XGX">(i'=i=«)2/2gr where the last two terms are the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done when the plane FIG. 154. OF WATER] HYDRAULICS 87 is moving in the same direction as the jet is Pu = (G/g). The former gives a minimum, the latter a maximum efficiency. Putting w = 311 in the expression above, ij max. =-/y. (3) If, instead of one plane moving before the jet, a scries of planes are introduced at short intervals at the same point, the quantity of water impinging on the series will be cot) instead of ta(v-u), and the whole pressure = (G/g)uv(v — u). The work done is (G/g)- (2M-t>)} = 2(G/g)co(t>-w)2. Comparing this with case 2, it is seen that the pressure on a hemispherical cup is double that on a flat plane. The work done on the cup=2(G/g)co (t>-«)2« foot- 2li-v FIG. 155. pounds per second. The efficiency of the jet is greatest when v = in that case the efficiency = 4?. If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is (»-«), then the whole pressure on the chain of cups is (G/g)a>r(r-(2«-r)j =2(G/g)av(v-u). In this case the efficiency is greatest when v = 2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cups. §157- (5) Caseofa FlatVane oblique to the Jet (fig.156). — Thiscase presents some difficulty. The water spreading on the plane in all FIG. 156. directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB=t> = velocity of water, AC =« = velocity of plane. Then, completing the parallelogram, AD represents in magnitude and direction the relative velocity of water and plane. Draw AE normal to the plane and DE parallel to the plane. Then the relative velocity AD may be regarded as con- sisting of two components, one AE normal, the other DE parallel to the plane. On the assumption that friction is insensible, DE is unaffected by impact, but AE is destroyed. Hence AE represents the entire change of velocity due to impact and the direction of that change. The pressure on the plane is in the direction AE, and its amount is = mass of water impinging per second X AE. Let DAE =9, and let AD =&y. Then AE =vr cos 6 ; DE =»r sin 8. If Q is the volume of water impinging on the plane per second, the change of momentum is (G/g)Qiv cos 0. Let AC = u=velocity of the plane, and let AC make the angle CAE=8 with the normal to the plane. The velocity of the plane in the direction AE = u cos S. The work of the jet on the plane = (G/g)Qfr cos 8 u cos 5. The same problem may be thus treated algebraically (fig. 157). Let BAF = a, and CAF =6. The velocity v of the water may be de- composed into AF=t> cos a normal to the plane, and FB=u sin a parallel to the plane. Similarly the velocity of the plane =u =AC = BD can be decomposed into BG = FE = M cos 5 normal to the plane, and DG = u sin 8 parallel to the plane. As friction is neglected, the velocity of the water parallel to the plane is unaffected by the im- pact, but its component v cos a normal to the plane becomes after impact the same as that of the plane, that is, u cos 5. Hence the change of velocity during impact = AE=» cos a-u cos S. The change of momentum per second, and consequently the normal d FIG. 157. pressure on the plane is N = (G/g) Q (v cos a-wcos 8). The pressure in the direction m which the plane is moving is P = N cos 8 = (G/g)Q (v cos a-u cos 8) cos 8, and the work done on the plane is PM = (G/g)Q(f cos a-u cos 5) u cos 5, which is the same expression as before, since AE =vr cos 8 —v cos a—u cos 8. In one second the plane moves so that the point A (fig. 158) comes to C, or from the position shown in full lines to the position shown in dotted lines. If the plane remained stationary, a length AB =w of the jet would impinge on the plane, but, since the plane moves in the same direction as the jet, only the length HB = AB-AH impinges, on the plane. But AH = AC cos SI cos o = u cos 8/ cos a, and therefore HB =v— u cos S/ cos o. Let oj = sectional area of jet ; volume impinging on plane p _ leR per second =Q=a(v-u cos 8/cos a)=u(i> cos a-u cos 8)/ cos a. Inserting this in the formulae above, we get B cos a-u cos 8)2; (v cos a-ttcos S)2; (i) (2) (3) cos o Three cases may be distinguished : — (o) The plane is at rest. Then «=o, N = (G/g)o>ti2 cos a; and the work done on the plane and the efficiency of the jet are zero. (6) The plane moves parallel to the jet. Then 8 = 0, and P« = (G/g) o>Mcos2a(ti—«)2, which is a maximum when u = \v. When U = \TI then Pu max. = ^?(G/g)wt)s cos 2a, and the efficiency = !? = j COS 2O. (c) The plane moves perpendicularly to the jet. Then 8 = go°-a; cos 5 = sin a; and PM= — uMSln a(v cos o-wsin o)2. This is a maxi- g cos a mum when « = |» cos a. When u = Ju cos o, the maximum work and the efficiency are the same as in the last case. § 158. Best Form of Vane to receive Water. — When water impinges normally or obliquely on a plane, it is scattered in all directions after impact, and the work carried away by the water is then gener- ally lost, from the impossibility of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane, FIG. 159. however, the water may be entirely deviated in one direction, and the loss of energy from agitation of the water is entirely avoided. Let AB (fig. 159) be a vane, on which a jet of water impinges at the point A and in the direction AC. Take AC =v = velocity of 88 HYDRAULICS [IMPACT AND REACTION water, and let AD represent in magnitude and direction the velocity of the vane. Completing the parallelogram, DC or AE represents the direction in which the water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it will glide up the vane in the direction AB. This is sometimes expressed by saying that the vane receives the water without shock, § 159. Floats of Poncelet Water Wheels. — Let AC (fig. 160) repre- sent the direction of a thin horizontal stream of water having the A, FIG. 160. ™ velocity v. Let AB be a curved float moving horizontally with velocity u. The relative motion of water and float is then initially horizontal, and equal to v — u. In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the float at A should be tangential to the direction AC of relative motion. At the end of (v— u)/g seconds the float moving with the velocity « comes to the position AiB,, and during this time a particle of water received at A and gliding up the float with the relative velocity v — u, attains a height DE = (»— «)2/2£. At E the water comes to relative rest. It then descends along the float, and when after 2(v—u)/g seconds the float has come to A2B2 the water will again have reached the lip at A2 and will quit it tangentially, that is, in the direction CA2, with a relative velocity — (» — «) = —V (2gDE) acquired under the influ- ence of gravity. The absolute velocity of the water leaving the float is therefore u — (v — u)=2u—v. If u = % v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle; the stream of water has considerable thickness (about 8 in.); in order to get the water into and out of the wheel, it is then necessary that the lip of the float should make a small angle (about 15°) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining. § 1 60. Pressure on a Curved Surface when the Water is deviated wholly in one Direction. — When a jet of water, impinges on a curved surface in such a direction that it is received without shock, the pressure on the surface is due to its gradual deviation from its first direction. On any portion of the area the pressure is equal and opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water. Case I. Surface Cylindrical and Stationary. — Let AB (fig. 161) be the surface, having its axis at O and its radius =r. Let the water impinge at A tangentially, and quit the surface tangentially at B. Since the surface is at rest, v is both the absolute velocity of the water and the velocity relatively to the surface, and this remains un- changed during contact with the surface, because the deviating force is at each point perpendicular to the direction of motion. The water is deviated through an angle BCD=AOB=<#>. Each particle of water of weight p exerts radially a centrifugal force pv*/rg. Let the thickness of the stream = / ft. Then the weight of water resting on Ib; and the normal pressure per unit of n = Gtv*/gr. The resultant of the radial pressures uni- formly distributed from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of a force of intensity = n, acting on the projection of AB on a plane perpendicular to the direction OC. The length of the chord AB = 2r sin % ; let 6 = breadth of the surface perpendicular to the plane of the figure. The resultant pressure on surface unit of surface FIG. 161. surface = G< 2 g r g which is independent of the radius of curvature. It may be inferred that the resultant pressure is the same for any curved surface of the same projected area, which deviates the water through the same angle. Case 2. Cylindrical Surface moving in the Direction AC with Velo- city u. — The relative velocity = v — u. The final velocity BF (fig. 162) is found by combining the relative velocity BD=zi — u tangential to the surface with the velocity BE = M of the surface. The intensity of normal pressure, as in the last case, is (G/g)t(v—u)2/r. The resultant FIG. 162. normalpressureR = 2(G/g)6/(i> — w)2sin J . This resultant pressure may be resolved into two components P and L.one parallel and the other perpendicular to the direction of the vane's motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work. P = R sin j* = (G/g)i/(u — M)S(I —cos*) ; The work done by the jet on the vane is Pu = (G/g)btu(v — tt)*(i- cos 4>), which is a maximum when u = \v. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected. If *=i8o°, cos <#>=— i, i— cos. — u.cot= 2; then P = 2(G/g)bt(v-u)', the same result as for a concave cup. § 161. Position which a Movable Plane takes in Flowing Water. — When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an in- definite current of fluid, it takes a position such that the resultant of the normal pres- sures on the two sides of the axis passes through the axis. If, therefore, planes pivoted so that the ratio a/b (fig. 163) is varied are placed in water, and the angle they make with the direction of the stream is observed, the position of the resultant of the pressures on FIG. 163. the plane is determined for different angular positions. Experiments of this kind have been made by Hagen. Some of his results are given in the following table: — Larger plane. Smaller Plane. a/b = I -o =... 0 = 90° 0-9 75° 72i° 0-8 60° 57° 07 48° 43° 0-6 25° 29° o-5 13° 13° 0-4 8° 6J° 0-3 6J° 0-2 4° § 162. Direct Action distinguished from Reaction (Rankine, Steam Engine, § 147). The pressure which a jet exerts on a vane can be distinguished into two parts, viz.: — (1) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. !53- § '54' ao c°s bae is the direct component of the water's velocity, or component in the direction of motion of vane. This is changed into the velocity ae of the vane. The pressure due to direct impulse is then PI =GQ(a6 cos bae—ae)/g. For a flat vane moving normally, this direct action is the only action producing pressure on the vane. (2) The term reaction is applied to the additional action due to the direction and velocity with which the water glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, § 160, the direct pressure is That due to reaction is Pa- - If 4><90°, the direct component of the water's motion is not wholly converted into the velocity of the vane, and the whole OF WATER] HYDRAULICS 89 pressure due to direct impulse is not obtained. If >go°, cos ^ is negative and an additional pressure due to reaction is obtained. § 163. Jet Propeller. — In the case of vessels propelled by a jet of water (fig. 164), driven stern wards from orifices at the side of the vessel, the water, originally at rest out- side the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected sternwards from the jets with a velocity v relatively to the ship, or ti— V relatively to the earth. If U is the total sectional area of the jets, Qv is the quantity of water discharged per second. The momentum generated per second in a sternward direction is u — V), and this is equal to the forward acting reaction P O FIG. 164. which propels the ship. The energy carried away by the water (I) (2) Adding (i) and (2), we get the whole work expended on the water, neglecting friction : — The useful work done on the ship Hence the efficiency of the jet propeller is PV/W=2V/(i>+V). (3) This increases towards unity as v approaches V. In other words, the less the velocity of the jets exceeds that of the ship, and there- fore the greater the area of the orifice of discharge, the greater is the efficiency of the propeller. In the " Waterwitch " v was about twice V. Hence in this case the theoretical efficiency of the propeller, friction neglected, was about f. § 164. Pressure of a Steady Stream in a Uniform Pipe on a Plane normal to the Direction of Motion. — Let CD (fig. 165) be a plane placed normally to the stream which, for simplicity, may be sup- posed to flow horizontally. The fluid filaments are deviated in front of the plane, form a contraction at AiAi, and converge again, leaving a mass of eddying water behind the plane. Suppose the section AoAo taken at a point where the parallel motion has not begun to be disturbed, and A2A2 where the parallel motion is re- established. Then since the same quantity of water with the same velocity passes AoAo, A2A2 in any given time, the external forces produce no change of momentum on the mass AoAoA2A2, and must therefore be in equilibrium. If Ji is the section of the stream at AoAo or A2A2, and o> the area of the plate CD, the area of the con- tracted section of the stream at AiAi will be cc(O— &>), where cc is the coefficient of contraction. Hence, if v is the velocity at AoAo or A2A2, and i>! the velocity at AiAi, ..ic. Let pa, pi, pi be the pressures at the three sections. Applying Bernoulli's theorem to the sections AoAo and AiAi, 2g Also, for the sections AiAi and A2A2, allowing that the head due to the relative velocity Vi— v is lost in shock: — Pi ,vf_p2 . f2 , fa -i')2 +-+ 2g or, introducing the value in (i), (2) (3) Now the external forces in the direction of motion acting on the mass AoA0A2A2 are the pressures p&l, , — p£l at the ends, and the reaction — R of the plane on the water, which is equal and opposite to the pressure of the water on the plane. As these are in equilibrium, (A) an expression like that for the pressure of an isolated jet on an indefinitely extended plane, with the addition of the term in brackets, which depends only on the areas of the stream and the plane. For a given plane, the expression in brackets diminishes as Q increases. If B/w = p, the equation (4) becomes which is of the form R=Go)(r2/2g)K, where K depends only on the ratio of the sections of the stream and plane. For example, let cc = o-85, a value which is probable, if we allow that the sides of the pipe act as internal borders to an orifice. Then I 2 3 4 5 10 50 IOO K = 00 3-66 1-75 1-29 I-IO •94 2-OO 3-50 The assumption that the coefficient of contraction c, is constant for different values of p is probably only true when p is not very large. Further, the increase of K for large values of p is contrary to experience, and hence it may be inferred that the assumption that all the filaments have a common velocity Hi at the section AiAi and a common velocity v at the section A2A2 is not true when the stream is very much larger than the plane. Hence, in the expression , Vi=vp/cc(p—l), t)2 = rp/(p — i). R = KiGu»2/2g, K must be determined by experiment in each special case. For a cylindrical body putting a for the section, cc for the coefficient of contraction, c»(J2— w) for the area of the stream at AiAi, or, putting Then where . Taking cc = o-8s and p = 4, Ki =0-467, a value less than before. Hence there is less pressure on the cylinder than on the thin plane. § 165. Distribution of Pressure on a Surface on which a Jet impinges normally. — The principle of momentum gives readily enough the total or resultant pressure of a jet impinging on a plane surface, but in some cases it is useful to know the distribution of the pressure. The problem in the case in which the plane is struck normally, and the jet spreads in all directions, is one of great complexity, but even in that case the maximum intensity of the pressure is easily assigned. Each layer of water flowing from an orifice is gradually deviated (fig. 1 66) by contact with the sur- face, and during deviation exercises a centrifugal pressure towards the _. axis of the jet. The force exerted by each small mass of water is normal to its path and inversely as FIG. 166. the radius of curvature of the path. Hence the greatest pressure on the plane must be at the axis of the jet, and the pressure must decrease from the axis outwards, in some such way as is shown by the curve of pressure in fig. 167, the branches of the curve being probably asymptotic to the plane. For simplicity suppose the jet is a vertical one. Let hi (fig. 167) be the depth of the orifice from the free surface, and fi the velocity of discharge. Then, if w is the area of the orifice, the quantity of water impinging on the plane is obviously Q=(d»i=a>V(2g/,i); that is, supposing the orifice rounded, and neglecting the coefficient of discharge. The velocity with which the fluid reaches the plane is, however, greater than this, and may reach the value where h is the depth of the plane below the free surface. The external layers of fluid subjected throughout, after leaving the orifice, to the atmospheric pressure will attain the velocity v, and will flow away with this velocity unchanged except by friction. The layers towards the interior of the jet, being subjected to a pressure greater than atmospheric pressure, will attain a less velocity, and so much less as they are nearer the centre of the jet. But the pressure 9o HYDRAULICS [IMPACT AND REACTION can in no case exceed the pressure iPJ2g or h measured in feet of water, or the direction of motion of the water would be reversed, and there would be reflux. Hence the maximum intensity of the pressure FIG. 167. of the jet on the plane is h ft. of water. If the pressure curve is drawn with pressures represented by feet of water, it will touch the free water surface at the centre of the jet. Suppose the pressure curve rotated so as to form a solid of revolu- tion. The weight of water contained in that solid is the total pressure of the jet on the surface, which has already been deter- mined. Let V = volume of this solid, then GV is its weight in pounds. Consequently GV = (G/g)wt»i»; V=2«V(AA,). We have already, therefore, two conditions to be satisfied by the pressure curve. Some very interesting experiments on the distribution of pressure on a surface struck by a jet have been made by J. S. Beresford (Prof. Papers on Indian Engineering, No. cccxxii.), with a view to afford information as to the forces acting on the aprons of weirs. Cylindrical jets i in. to 2 in. diameter, issuing from a vessel in which the water level was constant, were allowed to fall vertically on a brass plate 9 in. in diameter. A small hole in the brass plate communicated by a flexible tube with a vertical pressure column. Arrangements were made by which this aperture could be moved fa in. at a time across the area struck by the jet. The height of the pressure column, for each position of the aperture, gave the pressure at that point of the area struck by the jet. When the aperture was o os 1-6 Distance from axis of let ID inches. FIG. 1 68. — Curves of Pressure of Jets impinging normally on a Plane. exactly in the axis of the jet, the pressure column was very nearly level with the free surface in the reservoir supplying the jet ; that is, the pressure was very nearly n2/2g. As the aperture moved away from the axis of the jet, the pressure diminished, and it became insensibly small at a distance from the axis of the jet about equal to the dia- meter of the jet. Hence, roughly, the pressure due to the jet extends over an area about four times the area of section of the jet. Fig. 168 shows the pressure curves obtained in three experiments with three jets of the sizes shown, and with the free surface level in the reservoir at the heights marked. Experiment i. Jet "475 in. diameter. Experiment 2. Jet '988 in. diameter. Experiment 3. Jet 19' 5 in. diameter. llj •a . *** 1 Si .9 il 1 ||j •~ Ib per sq. ft., where G is the weight ofa cubic foot of air and » the velocity of the current in ft. per sec. On the leeward side the negative pressure is uniform except near the edges, and its value depends on the form of the plate. For a circular plate the pressure on the leeward side was 0-48 Gr2/2g and for a rectangular plate 0-66 Gi>2/2g. For circular or square plates the resultant pressure on the plate was P =0-00126 if ft per sq. ft. where v is the velocity of the current in ft. per sec. On a long narrow rectangular plate the resultant pressure was nearly 60% greater than on a circular plate. In later tests on larger planes in free air, Stanton found resistances 18% greater than those observed with small planes in the air trunk. § 168. Case when the Direction of Motion is oblique to the Plane. — The determination of the pressure between a fluid and surface in this case is of importance in many practical questions, for instance, in assigning the load due to wind pressure on sloping and curved roofs, and experiments have been made by Hutton, Vince, and Thibault on planes moved circularly through air and water on a whirling machine. FIG. 169. Let AB (fig. 170) be a plane moving in the direction R makin an angle with the plane. The resultant pressure between the fluii and the plane will be a normal pressure N. The component R of this normal pressure is the resistance to the motion of the plane and the other component L is a lateral force resisted by the guides which support the plane. Obviously R = N sin ; L = N cos <£. In the case of wind pressure on a sloping roof surface, R is the horizontal and L the vertical component of the normal pres- N FIG. 170. In experiments with the whirling machine it is the resistance to motion, R, which is directly measured. Let P be the pressure on a plane moved normally through a fluid. Then, for the same plane inclined at an angle to its direction of motion, the resistance was found by Hutton to be R = P(sin 4.) 1-842 cos*. A simpler and more convenient expression given by Colonel Duchemin is Consequently, the total pressure between the fluid and plane is N =2? sin /(i +sin2 ) =2P/(cosec + sin <#>), and the lateral force is L = 2P sin 0 cos <£/(i -fsin2 ). In 1872 some experiments were made for the Aeronautical Society on the pressure of air on oblique planes. These plates, of I to 2 ft. square, were balanced by ingenious mechanism designed by F. H. Wenham and Spencer Browning, in such a manner that both the pressure in the direction of the air current and the lateral force were separately measured. These planes were placed opposite a blast from a fan issuing from a wooden pipe 18 in. square. The pressure of the blast varied from -ft to I in. of water pressure. The following are thejresults given in pounds per square foot of the plane, and a com- parison of the experimental results with the pressures given by Duchemin's rule. These last values are obtained by .taking P =3-31, the observed pressure on a normal surface :— Angle between Plane and Direction ) of Blast \ 15° 20° 60° 90° Horizontal pressure R . . . . Lateral pressure L 0-4 1-6 1-65 1-605 0-61 1-96 2-05 2-027 273 1-26 3-01 3-276 3-3i 3-31 3-31 Normal pressure VL2 + R2 . Normal pressure by Duchemin's rule WATER MOTORS In every system of machinery deriving energy from a natural water-fall there exist the following parts: — 1. A supply channel or head race, leading the water from the highest accessible level to the site of the machine. This may be an open channel of earth, masonry or wood, laid at as small a slope as is consistent with the delivery of the necessary supply of water, or it may be a closed cast or wrought-iron pipe, laid at the natural slope of the ground, and about 3 ft. below the surface. In some cases part of the head race is an open channel, part a closed pipe. The channel often starts from a small storage reservoir, constructed near the stream supplying the water motor, in which the water accumulates when the motor is not working. There are sluices or penstocks by which the supply can be cut off when necessary. 2. Leading from the motor there is a tail race, culvert, or discharge pipe delivering the water after it has done its work at the lowest convenient level. 3. A waste channel, weir, or bye- wash is placed at the origin of the head race, by which surplus water, in floods, escapes. 4. The motor itself, of one of the kinds to be described presently, which either overcomes a useful resistance directly, as in the case of a ram acting on a lift or crane chain, or indirectly by actuating transmissive machinery, as when a turbine drives the shafting, belting and gearing of a mill. With the motor is usually com- bined regulating machinery for adjusting the power and speed to the work done. This may be controlled in some cases by automatic governing machinery. HYDRAULICS [WATER MOTORS § 169. Water Motors with Artificial Sources of Energy. — The great convenience and simplicity of water motors has led to their adoption in certain cases, where no natural source of water power is available. In these cases, an artificial source of water power is created by using a steam-engine to pump water to a reservoir at a great elevation, or to pump water into a closed reservoir in which there is great pressure. The water flowing from the reservoir through hydraulic engines gives back the energy expended, less so much as has been wasted by friction. Such arrangements are most useful where a continuously acting steam engine stores up energy by pumping the water, while the work done by the hydraulic engines is done intermittently. § 170. Energy of a- Water-fall. — Let H, be the total fall of level from the point where the water is taken from a natural stream to the point where it is discharged into it again. Of this total fall a portion, which can be estimated independently, is expended in overcoming the resistances of the head and tail races or the supply and discharge pipes. Let this portion of head wasted be I),.. Then the available head to work the motor is H =Hi — f)r. It is this available head which should be used in all calculations of the proportions of the motor. Let Q be the supply of water per second. Then GQH foot-pounds per second is the gross available work of the fall. The power of the fall may be utilized in three ways, (a) The GQ pounds of water may be placed on a machine at the highest level, and descending in con- tact with it a distance of H ft., the work done will be (neglecting losses from friction or leakage) GQH foot-pounds per second. (6) Or the water may descend in a closed pipe from the higher to the lower level, in which case, with the same reservation as before, the pressure at the foot of the pipe will be p = GH pounds per square foot. If the water with this pressure acts on a movable piston like that of a steam engine, it will drive the piston so that the volume described is Q cubic feet per second. Then the work done will be pQ = GHQ foot-pounds per second as before, (c) Or lastly, the water may be allowed to acquire the velocity v = V 2gH by its descent. The kinetic energy of Q cubic feet will then be iGOy/g = GQH, and if the water is allowed to impinge on surfaces suitably curved which bring it finally to rest, it will impart to these the same energy as in the previous cases. Motors which receive energy mainly in the three ways described in (a), (b), (c) may be termed gravity, pressure and inertia motors respectively. Generally, if Q ft. per second of water act by weight through a distance hi, at a pressure p due to hi ft. of fall, and with a velocity v due to h, ft. of fall, so that hi+h^+h3 = H, then, apart from energy wasted by friction or leakage or imperfection of the machine, the work done will be GQA+pQ+(G/g)Q(W2g)=GQH foot pounds, the same as if the water acted simply by its weight while descending H ft. § 171. Site for Water Motor. — Wherever a stream flows from a higher to a lower level it is possible to erect a water motor. The amount of power obtainable depends on the available head and the supply of water. In choosing a site the engineer will select a portion of the stream where there is an abrupt natural fall, or at least a considerable slope of the bed. He will have regard to the facility of constructing the channels which are to convey the water, and will take advantage of any bend in the river which enables him to shorten them. He will have accurate measurements made of the quantity of water flowing in the stream, and he will endeavour to ascertain the average quantity available throughout the year, the minimum quantity in dry seasons, and the maximum for which bye-wash channels must be provided. In many cases the natural fall can be increased by a dam or weir thrown across the stream. The engineer will also examine to what extent the head will vary in different seasons, and whether it is necessary to sacrifice part of the fall and give a steep slope to the tail race to prevent the motor being drowned by backwater in floods. Streams fed from lakes which form natural reservoirs or fed from glaciers are less variable than streams depending directly on rainfall, and are therefore advan- tageous for water-power purposes. i 172. Water Power at Holyoke, U.S.A. — About 85 m. from the mouth of the Connecticut river there was a fall of about 60 ft. in a short distance, forming what were called the Grand Rapids, below which the river turned sharply, forming a kind of peninsula on which the city of Holyoke is built. In 1845 the magnitude of the water- power available attracted attention, and it was decided to build a dam across the river. The ordinary flow of the river is 6000 cub. ft. per sec., giving a gross power of 30,000 h.p. In dry seasons the power is 20,000 h.p., or occasionally less. From above the dam a system of canals takes the water to mills on three levels. The first canal starts with a width of 140 ft. and depth of 22 ft., and supplies the highest range of mills. A second canal takes the water which has driven turbines in the highest mills and supplies it to a second series of mills. There is a third canal on a still lower level supplying the lowest mills. The water then finds its way back to the river. With the grant of a mill site is also leased the right to use the water- power. A mill-power is defined as 38 cub. ft. of water per sec. during 16 hours per day on a fall of 20 ft. This gives about 60 h.p. effective. The charge for the power water is at the rate of 2os. per h.p. per annum. § 173. Action of Water in a Water Motor. — Water motors may be divided into water-pressure engines, water-wheels and turbines. Water-pressure engines are machines with a cylinder and piston or ram, in principle identical with the corresponding part of a steam-engine. The water is alternately admitted to and dis- charged from the cylinder, causing a reciprocating action of the piston or plunger. It is admitted at a high pressure and dis- charged at a low one, and consequently work is done on the piston. The water in these machines never acquires a high velocity, and for the most part the kinetic energy of the water is wasted. The useful work is due to the difference of the pressure of admission and discharge, whether that pressure is due to the weight of a column of water of more or less considerable height, or is artificially produced in ways to be described presently. Water-wheels are large vertical wheels driven by water falling from a higher to a lower level. In most water-wheels, the water acts directly by its weight loading one side of the wheel and so causing rotation. But in all water-wheels a portion, and in some a considerable portion, of the work due to gravity is first em- ployed to generate kinetic energy in the water; during its action on the water-wheel the velocity of the water diminishes, and the wheel is therefore in part driven by the impulse due to the change of the water's momentum. Water-wheels are there- fore motors on which the water acts, partly by weight, partly by impulse. Turbines are wheels, generally of small size compared with water wheels, driven chiefly by the impulse of the water. Before entering the moving part of the turbine, the water is allowed to acquire a considerable velocity; during its action on the turbine this velocity is diminished, and the impulse due to the change of momentum drives the turbine. In designing or selecting a water motor it is not sufficient to consider only its efficiency in normal conditions of I working. It is generally quite as important to know how it will act with a scanty water supply or a diminished head. The greatest difference in water motors is in their adaptability to varying conditions of working. Water-pressure Engines. §174. In these the water acts by pressure either due to the height of the column in a supply pipe descending from a high- level reservoir, or created by pumping. Pressure engines were first used in mine-pumping on waterfalls of greater height than could at that time be utilized by water wheels. Usually they were single acting, the water-pressure lifting the heavy pump rods which then made the return or pumping stroke by their own weight. To avoid losses by fluid friction and shock the velocity of the water in the pipes and passages was restricted to from 3 to 10 ft. per second, and the mean speed of plunger to i ft. per second. The stroke was long and the number of strokes 3 to 6 per minute. The pumping lift being constant, such engines worked practically always at full load, and the efficiency was high, about 84%. But they were cumbrous machines. They are described in Weisbach's Mechanics of Engineering. The convenience of distributing energy from a central station to scattered working-points by pressure water conveyed in pipes — a system invented by Lord Armstrong — has already been mentioned. This system has led to the development of a great variety of hydraulic pressure engines of very various types. The cost of pumping the pressure water to some extent restricts its use to intermittent operations, such as working lifts and cranes, punching, shearing and riveting machines, forging and flanging presses. To keep down the cost of the distributing WATER MOTORS] HYDRAULICS generally 700 ft per 93 mains very high pressures are adopted, sq. in. or 1600 ft. of head or more. In a large number of hydraulic machines worked by water at high pressure, especially lifting machines, the motor consists of a direct, single acting ram and cylinder. In a few cases double- acting pistons and cylinders are used; but they involve a water-tight packing of the piston not easily accessible. In some cases pressure engines are used to obtain rotative movement, and then two double-acting cylinders or three single-acting cylinders are used, driving a crank shaft. Some double-acting cylinders have a piston rod half the area of the piston. The pressure water acts continuously on the annular area in front of the piston. During the forward stroke the pressure on the front of the piston balances half the pressure on the back. During the return stroke the pressure on the front is unopposed. The water in front of the piston is not exhausted, but returns to the supply pipe. As the frictional losses in a fluid are independent of the pressure, and the work done increases directly as the pressure, the percentage loss decreases for given velocities of flow as the pressure increases. Hence for .high-pressure machines somewhat greater velocities are permitted in the passages than for low-pressure machines. In supply mains the velocity is from 3 to 6 ft. per second, in valve passages 5 to 10 ft. per second, or in extreme cases 20 ft. per second, where there is less object in economizing energy. As the water is incompressible, slide valves must have neither lap nor lead, and piston valves are preferable to ordinary slide valves. To prevent injurious com- pression from exhaust valves closing too soon in rotative engines with a fixed stroke, small self-acting relief valves are fitted to the cylinder ends, opening outwards against the pressure into the valve chest. Imprisoned water can then escape without over- straining the machines. In direct single-acting lift machines, in which the stroke is fixed, and in rotative machines at constant speed it is obvious that the cylinder must be filled at each stroke irrespective of the amount of work to be done. The same amount of water is used whether much or little work is done, or whether great or small weights are lifted. Hence while pressure engines are very efficient at full load, their efficiency decreases as the load de- creases. Various arrangements have been adopted to diminish this defect in engines working with a variable load. In lifting machinery there is sometimes a double ram, a hollow ram enclosing a solid ram. By simple arrangements the solid ram only is used for small loads, but for large loads the hollow ram is locked to the solid ram, and the two act as a ram of larger area. In rotative engines the case is more difficult. In Hastie's and Rigg's engines the stroke is automatically varied with the load, increasing when the load is large and decreasing when it is small. But such engines are complicated and have not achieved much success. Where pressure engines are used simplicity is generally a first consideration, and economy is of less importance. § 175. Efficiency of Pressure Engines. — It is hardly possible to form a theoretical expression for the efficiency of pressure engines, but some general considerations are useful. Consider the case of a' long stroke hydraulic ram, which has a fairly constant velocity v during the stroke, and valves which are fairly wide open during most of the stroke. Let r be the ratio of area of ram to area of valve passage, a ratio which may vary in ordinary cases from 4 to 12. Then the loss in shock of the water entering the cylinder will be (r— l)V/2g in ft. of head. The friction in the supply pipe is also proportional to v1. The energy carried away in exhaust will be proportional to i>2. Hence the total hydraulic losses may be taken to be approximately jT!/2g ft., where f is a coefficient depending on the proportions of the machine. Let / be the friction of the ram packing and mechanism reckoned in ft per sq. ft. of ram area. Then if the supply-pipe pressure driving the machine is p ft per sq. ft., the effective working pressure will be p-G£v*/2g-f ft per sq. ft. Let A be the area of the ram in sq. ft., v its velocity in ft. per sec. The useful work done will be (p-G?v2/2g-f)Av ft. ft per sec., and the efficiency of the machine will be regulating the engine for varying load the pressure is throttled, part of the available head is destroyed at the throttle valve, and p in the bracket above is reduced. Direct-acting hydraulic lifts, without intermediate gearing, may have an efficiency of 95 % during the working stroke. If a hydraulic jigger is used with ropes and sheaves to change the speed of the ram to the speed 'of the lift, the efficiency may be only 50%. E. B. Ellington has given the efficiency of lifts with hydraulic balance at 85% during the working stroke. Large pressure engines have an efficiency of 85 %, but small rota- tive engines probably not more than 50 % and that only when fully loaded. Level of This shows that the efficiency increases with the pressure p, and diminishes with the speed », other things being the same. If in § 176. Direct-Acting Hydraulic Lift (fig. 171).— This is the simplest of all kinds of hydraulic motor. A cage W is lifted directly by water pressure acting in a cylinder C, the length of which is a little greater than the lift. A ram or plunger R of the same length is attached to the cage. The water-pressure admitted by a cock to the cylinder forces up the ram, and when the supply valve is closed and the discharge valve opened, the ram descends. In this case the ram is 9 in. diameter, with a stroke of 49 ft. It consists of lengths of wrought-iron pipe screwed together perfectly water- tight, the lower end being closed by a cast-iron plug. The ram works in a cylinder n in. dia- meter of 9 ft. lengths of flanged cast-iron pipe. The ram passes water-tight through the cylinder cover, which is provided with double hat leathers to prevent leakage outwards or inwards. As the weight of the ram and cage is much more than sufficient to cause a descent of the cage, part of the weight is balanced. A chain at- tached to the cage passes over a pulley at the top of the lift, and carries at its free end a balance weight B, working in f iron guides. Water is ad- mitted to the cylinder from a 4-in. supply pipe through a two- way slide, worked by a rack, spindle and endless rope. The lift works under 73 ft. of head, and lifts 1350 ft at 2 ft. per second. The effi- ciency is from 75 to 80%. The principal pre- judicial resistance to the motion of a ram of this kind is the fric- tion of the cup leathers, which make the joint between the cylinder and ram. Some ex- "°- I7I- periments by John Hick give for the friction of these the following formula. Let F= the total friction in W H leathers pounds; 94 HYDRAULICS [WATER MOTORS d = diameter of ram in ft.; p = water-pressure in pounds per sq. ft.; k a coefficient. F = kpd £ = 0-00393 if the leathers are new or badly lubricated; = 0-00262 if the leathers are in good condition and well lubricated. Since the total pressure on the ram is P = \TrcPp, the fraction of the total pressure expended in overcoming the friction of the leathers is F/P = -005/d to -0033/d, d being in feet. Let H be the height of the pressure column measured from the free surface of the supply reservoir to the bottom of the ram in its lowest position, Hi the height from the discharge reservoir to the same point, h the height of the ram above its lowest point at any moment, S the length of stroke, JJ the area of the ram, W the weight of cage, R the weight of ram, B the weight of balance weight, w the weight of balance chain per foot run, F the friction of the cup leather and slides. Then, .neglecting fluid friction, if the ram is rising the accelerating force is and if the ram is descending If iu = % Gtt, PI and Pz are constant throughout the stroke; and the moving force in ascending and descending is the same, if B = W+R+a;S-Gn(H+H6)/2. Using the values just found for w and B, Let W+R+a>S+B = U, and let P be the constant accelerating force acting on the system, then the acceleration is (P/U)g. The velocity at the end of the stroke is (assuming the friction to be constant) r = V(2PgS/U); and the mean velocity of ascent is |u. § 177. Armstrong's Hydraulic Jigger. — This is simply a single- acting hydraulic cylinder and ram, provided with sheaves so as to give motion to a wire rope or chain. It is used in various forms of lift and crane. Fig. 172 shows the arrangement. A hydraulic ram or plunger B works in a stationary cylinder A. Ram and cylinder carry sets of sheaves over which passes a chain or rope, fixed at one end to the cylinder, and at the other connected over guide pulleys to a lift or crane. For each pair of pulleys, one on the cylinder and one on the ram, the movement of the free end of the rope is doubled compared with that of the ram. With three pairs of pulleys the free end of the rope has a movement equal \ to six times the stroke of the ram, the force i exerted being in the inverse proportion. \ § 178. Rotative Hydraulic Engines. — Valve- gear mechanism similar in principle to that of steam engines can be applied to actuate the admission and discharge valves, and the pressure engine is then converted into a con- tinuously-acting motor. Let H be the available fall to work the engine after deducting the loss of head in the supply and discharge pipes, Q the supply of water in cubic feet per second, and ij the efficiency of the engine. Then the horse-power of the engine is H.P.=,GQH/s5o. The efficiency of large slow-moving pressure engines is >/= -66 to -8. In small motors of this kind probably ij is not greater than -5. Let v be the mean velocity of the piston, then its diameter d is given by the relation Q = Trd*v/A in double-acting engines, = ir .=o-75(GQH/55o) =0-085 QH. If the peripheral velocity of the water wheel is too great, water is thrown out of the buckets before reaching the bottom of the fall. In practice, the circumferential velocity of water wheels of the kind now described is from 4^ to 10 ft. per second, about 6 ft. being the usual velocity of good iron wheels not of very small size. In order that the water may enter the buckets easily, it must have a greater velocity than the wheel. Usually the velocity of the water at the point where it enters the wheel is from 9 to 12 ft. per second, and to produce this it must enter the wheel at a point 16 to 27 in. below the head-water level. Hence the diameter of an overshot wheel may be D = H-iJtoH-2ift. Overshot and high breast wheels work badly in back-water, and hence if the tail-water level varies, it is better to reduce the diameter of the wheel so that its greatest immersion in flood is not more than i ft. The depth d of the shrouds is about 10 to 16 in. The number of buckets may be about N = jrD/<2. Let v be the peripheral velocity of the wheel. Then the capacity of that portion of the wheel which passes the sluice in one second is = » b d nearly, b being the breadth of the wheel between the shrouds. If, however, this quantity of water were allowed to pass on to the wheel the buckets would begin to spill their contents almost at the top of the fall. To diminish the loss from spilling, it is not only necessary to give the buckets a suitable form, but to restrict the water supply to one-fourth or one-third of the gross bucket capacity. Let m be the value of this ratio; then, Q being the supply of water per second, Q = mQi = mbdv. This gives the breadth of the wheel if the water supply is known. The form of the buckets should be determined thus. The outer element of the bucket should be in the direction of motion of the water entering relatively to the wheel, so that the water may enter without splashing or shock. The buckets should retain the water as long as possible, and the width of opening of the buckets should be 2 or 3 in. greater than the thickness of the sheet of water entering. For a wooden bucket (fig. 180, A), take ab = distance between two buckets on periphery of wheel. Make ed = J eb, and 6c = J to j ab. Join cd. For an iron bucket (fig. 180, B), take ed = J eb; bc = Draw cO making an angle of 10° to 15° with the radius at c. On Oc take a centre giving a circular arc passing near d, and round the curve into the radial part of the bucket de. There are two ways in which the power of a water wheel is given off to the machinery driven. In wooden wheels and wheels with rigid arms, a spur or bevil wheel keyed on the axle of the , turbine will transmit the power to the shafting. It is obvious that the whole turning moment due to the weight of the water is then trans- mitted through the arms and axle of the water wheel. When the water wheel is an iron one, it usually has light iron suspension arms incapable of resisting the bending action due to the transmission of the turning effort to the axle. In that case spur segments are bolted to one of the shrouds, and the pinion to which the power is transmitted is placed so that the teeth in gear are, as nearly as may be, on the line of action of the resultant of the weight of the water in the loaded arc of the wheel. The largest high breast wheels ever constructed were probably the four wheels, each 50 ft. in diameter, and of 125 h.p., erected by Sir W. Fairbairn in 1825 at Catrine in Ayrshire. These wheels are still working. § 181. Poncdet Water Wheel. — When the fall does not exceed 6 ft., the best water motor to adopt in many cases is the Poncelet undershot water wheel. In this the water acts very nearly in the same way as in a turbine, and the Poncelet wheel, although slightly less efficient than the best turbines, in normal conditions of working, is superior to most of them when working with a reduced supply of water. A general notion of the action of the water on a Poncelet wheel has already been given in § 159. Fig. 181 shows its construction. The water penned back between the side walls of the wheel pit is allowed to flow to the FIG. 181. wheel under a movable sluice, at a velocity nearly equal to the velocity due to the whole fall. The water is guided down a slope of i in 10, or a curved race, and enters the wheel without shock. Gliding up the curved floats it comes to rest, falls back, and acquires at the point of discharge a backward velocity relative to the wheel nearly equal to the forward velocity of the wheel. Consequently it leaves the wheel deprived of nearly the whole of its original kinetic energy. Taking the efficiency at 0-60, and putting H for the available fall, h.p. for the horse-power, and Q for the water supply per second, h.p. = 0-068 QH. The diameter D of the wheel may be taken arbitrarily. It should not be less than twice the fall and is more often four times the fall. For ordinary cases the smallest convenient diameter is 14 ft. with a straight, or 10 ft. with a curved, approach channel. The radial TURBINES] HYDRAULICS 97 depth of bucket should be at least half the fall, and radius of curvature of buckets about half the radius of the wheel. The shrouds are usually of cast iron with flanges to receive the buckets. The buckets may be of iron J in. thick bolted to the flanges with f6 in. bolts. Let H' be the fall measured from the free surface of the head- water to the point F where the mean layer enters the wheel ; then the velocity at which the water enters is » = V (2gH'), and the best circumferential velocity of the wheel is V = 0-551; to o-6i>. The number of rotations of the wheel per second is N = V/irD. The thickness of the sheet of water entering the wheel is very im- portant. The best thickness according to experiment is 8 to 10 in. The maximum thickness should not exceed 12 to 15 in., when there is a surplus water supply. Let e be the thickness of the sheet of water entering the wheel, and b its width ; then bev = Q ; or b = Q/ev. Grashof takes e = JH, and then 6 = 6Q/HV(2gH). Allowing for the contraction of the stream, the area of opening through the sluice may be 1-25 be to 1-3 be. The inside width of the wheel is made about 4 in. greater than b. Several constructions have been given for the floats of Poncelet wheels. One of the simplest is that shown in figs. 181, 182. Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD making angles of 15° with OA. Then BD may be the length of FIG. 182. the close breasting fitted to the wheel. Draw the bottom of the head race BC at a slope of I in 10. Parallel to this, at distances je and e, draw EF and GH. Then EF is the mean layer and GH the surface layer entering the wheel. Join OF, and make OFK = 23°. Take FK=o-5 to 0-7 H. Then K is the centre from which the bucket curve is struck and KF is the radius. The depth of the shrouds must be sufficient to prevent the water from rising over the top of the float. It is £H to §H. The number of buckets is not very important. They are usually I ft. apart on the circumference of the wheel. The efficiency of a Poncelet wheel has been found in experiments to reach 0-68. It is better to take it at 0-6 in estimating the power of the wheel, so as to allow some margin. In fig. 182 Vi is the initial and va the final velocity of the water, v, parallel to the vane the relative velocity of the water and wheel, and V the velocity of the wheel. Turbines. § 182. The name turbine was originally given in France to any water motor which revolved in a horizontal plane, the axis being vertical. The rapid development of this class of motors dates from 1827, when a prize was offered by the Societe d'Encouragement for a motor of this kind, which should be an improvement on certain wheels then in use. The prize was ultimately awarded to Benoit Fourneyron (1802-1867), whose turbine, but little modified, is still constructed. Classification of Turbines. — In some turbines the whole available energy of the water is converted into kinetic energy before the water acts on the moving part of the turbine. Such turbines are termed Impulse or Action Turbines, and they are distinguished by this that the wheel passages are never entirely filled by the water. To ensure this condition they must be placed a little above the tail water and discharge into free air. Turbines in which part only of the available energy is converted into kinetic energy before the water enters the wheel are termed Pressure or Reaction Turbines. In these there is a pressure which in some cases amounts to half the head in the clearance space between the guide vanes and wheel vanes. The velocity with which the water enters the wheel is due to the difference between the pressure due to the head and the pressure in the clearance space. In pressure turbines the wheel passages must be continuously filled with water for good efficiency, and the wheel may be and generally is placed below the tail water level. Some turbines are designed to act normally as impulse turbines discharging above the tail water level. But the passages are so designed that they are just filled by the water. If the tail water rises and drowns the turbine they become pressure turbines with a small clearance pressure, but the efficiency is not much affected. Such turbines are termed Limit turbines. Next there is a difference of constructive arrangement of turbines, which does not very essentially alter the mode of action of the water. In axial flow or so-called parallel flow turbines, the water enters and leaves the turbine in a direction parallel to the axis of rotation, and the paths of the molecules lie on cylindrical surfaces concentric with that axis. In radial outward and inward flow turbines, the water enters and leaves the turbine in directions normal to the axis of rotation, and the paths of the molecules lie exactly or nearly in planes normal to the axis of rotation. In outward flow turbines the general direction of flow is away from the axis, and in inward flow turbines towards the axis. There are also mixed flow turbines in which the water enters normally and is discharged parallel to the axis of rotation. Another difference of construction is this, that the water may be admitted equally to every part of the circumference of the turbine wheel or to a portion of the circumference only. In the former case, the condition of the wheel passages is always the same; they receive water equally in all positions during rotation. In the latter case, they receive water during a part of the rotation only. The former may be termed turbines with complete admission, the latter turbines with partial admission. A reaction turbine should always have complete admission. An impulse turbine may have complete or partial admission. When two turbine wheels similarly constructed are placed on the same axis, in order to balance the pressures and diminish journal friction, the arrangement may be termed a twin turbine. If the water, having acted on one turbine wheel, is then passed through a second on the same axis, the arrangement may be termed a compound turbine. The object of such an arrangement would be to diminish the speed of rotation. Many forms of reaction turbine may be placed at any height not exceeding 30 ft. above the tail water. They then discharge into an air-tight suction pipe. The weight of the column of water in this pipe balances part of the atmospheric pressure, and the difference of pressure, producing the flow through the turbine, is the same as if the turbine were placed at the bottom of the fall. I. Impulse Turbines. (Wheel passages not filled, and discharging above the tail water.) [a) Complete admission. (Rare.) (b) Partial admission. (Usual.) II. Reaction Turbines. (Wheel passages filled, discharg- ing above or below the tail water or into a suction-pipe.) Always with complete admis- Axial flow, outward flow, inward flow, or mixed flow. Simple turbines; twin turbines; compound turbines. § 183. The Simple Reaction Wheel.— It has been shown, in § 162, that, when water issues from a vessel, there is a reaction on the vessel tending to cause motion in a direction opposite to that of the jet. This principle was applied in a rotating water motor at a very early period, and the Scotch turbine, at one time much used, differs in no essential respect from the older form of reaction wheel. The old reaction wheel consisted of a vertical pipe balanced on a vertical axis, and supplied with water (fig. 183). From the bottom of the vertical pipe two or more hollow horizontal arms extended, at the ends of which were orifices from which the water was dis- charged. The reaction of the jets caused the rotation of the machine. Let H be the available fall measured from the level of the water in the ver- tical pipe to the centres of the orifices, r the radius from the axis of rotation to the centres of the orifices, v the velocity of discharge through the jets, a the angular velocity of FIG. 183. XIV. 4 9» HYDRAULICS [TURBINES the machine. When the machine is at rest the water issues from the orifices with the velocity V (2gH) (friction being neglected). But when the machine rotates the water in the arms rotates also, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is H+aV2/2g ft. of water, and the velocity of discharge through the orifices is » = V (2gH-j-aV2). If the total area of the orifices is w, the quantity discharged from the wheel per second is Q = ui>=uV (2gH+a2r2). While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity ar. The absolute velocity of the water is therefore »- ar = V (2gH+aV2)-ar. The momentum generated per second is (GQ/g)(v-ar), which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore (GQ/g)(v-ar~)ar foot-pounds per sec. The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is _(v-ar) ar_ Let then ij = i-gH/2or+... which increases towards the limit I as ar increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when ar = V (2gH). Then the efficiency apart from friction is i, = (V(2aV)-or}ar/gH ' =o-4i4aV/gH =0-828, about 17 % of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60 %, so that about 21% of the energy of the fall is lost in friction, in addition to the energy carried away by the water. | 184. General Statement of Hydrodynamical Principles necessary for the Theory of Turbines. (a) When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation be- tween the changes of pressure and velocity is given by Bernoulli's theorem (§ 29). Suppose that, at a section A of such a passage, hi is the pressure measured in feet of water, »i the velocity, and Zi the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by fe, »2, z2. Then Al-&2 = W-f lJ)/2g +Z2-21. (l ) If the flow is horizontal, 22 = 21; and hi-hi = (f22-»i2)/2g. (la) (6) When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli's equation allowance must be made for the loss of head in shock (§ 36). Let vi, vi be the velocities before and after the abrupt change, then a stream of velocity »i impinges on a stream at a velocity vt, and the relative velocity is PI-VJ. The head lost is (»i-i>2)2/2g. Then equation (la) becomes To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curva- ture without discontinuity. (c) Equality of A ngular Impulse and Change of Angular Momen- tum.— Suppose that a couple, the moment of which is M, acts on a body of weight W for / seconds, during which it moves from Ai to A2 (fig. 184). Let »i be the velocity of the body at Ai, vt its velocity at A2, and let pi, pi be the perpendiculars from C on »i and t>2. Then M/ is termed the angular impulse of the couple, and W. the quantity FIG l8d and change of angular momentum is the change of angular momen- tum re'at'.ve'y to C. Then, from the equality of angular impulse or, if the change of momentum is estimated for one second, . M = Let n, r2 be the radii drawn from C to AI, A2, and let w,, Wi be the components of »i, r2, perpendicular to these radii, making angles ft and o with v\, fa. Then Wi sec ft ; t>2 =o>2 sec a ; = ri cos f);pi = rt cos a. /gXwr-zwi), (3) where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those points. Now the water flowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple -M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cub. ft. enter and leave the wheel per second, and let wi, wi be the tangential components of the velocity of the water at the receiv- ing and discharging surfaces of the wheel, ri, r? the radii of those surfaces By the principle above, - M =_(GQ/g) (wr-ovi). (4) If o is the angular velocity of the wheel, the work done by the water on the wheel is T = Ma= (GQ/g)(i0iri-i£>2r2)a foot-pounds per second. (5) § 185. Total and Available Fall. — Let H( be the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let t>p be that loss of head, which varies with the local conditions in which the turbine is placed. Then H=H,-hp is the available head for working the turbine, and on this the calcu- lations for the turbine should be based. In some cases it is necessary to place the turbine above the tail-water level, and there is then a fall J) from the centre of the outlet surface of the turbine to the tail- water level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H-Ij, but the efficiency of the turbine for the head H. § 1 86. Gross Efficiency and Hydraulic Efficiency of a Turbine. — Let Td be the useful work done by the turbine, in foot-pounds per second, Ti the work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local conditions in which the turbine is placed. Then the effective work done by the water in the turbine is The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is r,,=Td/GQH,. (6) And the hydraulic efficiency of the turbine alone is H-T/GQH._ (7)_ It is this last efficiency only with which the theory of turbines is concerned. From equations (5) and (7) we get TjGQH = (GQ/g) (a)iri-a»2r2)o ; r> = (iVirr^Wzrz) o/gH. (8) This is the fundamental equation in the theory of turbines. In general,1 Wi and w}, the tangential components of the water's motion on entering and leaving the wheel, are completely inde- pendent. That the efficiency may be as great as possible, it is obviously necessary that a>2 = o. In that case ij=aiiria/gH. (9) ori is the circumferential velocity of the wheel at the inlet surface. Calling this Vi, the equation becomes ij=i0iVj/gH. (90) This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Reiche (Turbinen- baues, 1877). § 187. General Description of a Reaction Turbine. — Professor James Thomson's inward flow or vortex turbine has been selected as the type of reaction turbines. It is one of the best in normal conditions of working, and the mode of regulation introduced is decidedly superior to that in most reaction turbines. Figs. 185 and 186 are external views of the turbine case; figs. 187 and 1 88 are the corresponding sections; fig. 189 is the turbine wheel. The example chosen for illustration has suction pipes, which permit the turbine to be placed above the tail-water level. The water enters the turbine by cast-iron supply pipes at A, and is discharged through two suction pipes S, S. The water 1 In general, because when the water leaves the turbine wheel it ceases to act on the machine. If deflecting vanes or a whirlpool are added to a turbine at the discharging side, then t»i may in part depend on i^, and the statement above is no longer true. TURBINES] HYDRAULICS 99 on entering the case distributes itself through a rectangular supply chamber SC, from which it finds its way equally to the four guide-blade passages G, G, G, G. In these passages it in equal proportions from each guide-blade passage. It consists of a centre plate /> (fig. 189) keyed on the shaft aa, which passes through stuffing boxes on the suction pipes. On each side of FIG. 185. FIG. 1 86. FIG. 187. acquires a velocity about equal to that due to half the fall, and is directed into the wheel at an angle of about 10° or 12° with the tangent to its circumference. The wheel W receives the water FIG. 188. the centre plate are the curved wheel vanes, on which the pressure of the water acts, and the vanes are bounded on each side by dished or conical cover plates c, c. Joint-rings j, j on the cover 100 HYDRAULICS [TURBINES plates make a sufficiently water-tight joint with the casing, to prevent leakage from the guide-blade chamber into the suction pipes. The pressure near the joint rings is not very great, probably not one-fourth the total head. The wheel vanes receive the water without shock, and deliver it into central spaces, from which it flows on either side to the suction pipes. The mode of regu- lating the power of the turbine is very simple. The guide- blades are pivoted to the case at their inner ends, and they are connected by a link- work, so that they all open and close simul- taneously and equally. In this way the area of opening through the guide- blades is altered with- out materially alter- ing the angle or the other conditions of the delivery into the wheel. The guide- blade gear may be FIG. 189. variously arranged. In this example four spindles, passing through the case, are linked to the guide- blades inside the case, and connected together by the links - • FIG. 190. I, I, I on the outside of the case. A worm wheel on one of the spindles is rotated by a worm d, the motion being thus slow enough to adjust the guide-blades very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal. Fig. 190 shows another arrangement of a similar turbine, with some adjuncts not shown in the other drawings. In this case the turbine rotates horizontally, and the turbine case is placed entirely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pentrough, containing a strainer, which prevents sticks and other solid bodies getting into the turbine. The turbine rests on three foundation stones, and, the pivot for the vertical shaft being under water, there is a screw and lever arrange- ment for adjusting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the right are the worm and wheel for working the guide-blade gear. § 1 88. Hydraulic Power at Niagara. — The largest development of hydraulic power is that at Niagara. The Niagara Falls Power Company have constructed two power houses on the United States side, the first with 10 turbines of 5000 h.p. each, and the second with 10 turbines of 5500 h.p. The effective fall is 136 to 140 ft. In the first power house the turbines are twin outward flow reaction turbines with vertical shafts running at 250 revs, per minute and driving the dynamos direct. In the second power house the turbines FIG. 191. are inward flow turbines with draft tubes or suction pipes. Fig. 191 shows a section of one of these turbines. There is a balancing piston keyed on the shaft, to the under side of which the pressure due to the fall is admitted, so that the weight of turbine, vertical shaft and part of the dynamo is water borne. About 70,000 h.p. is daily distributed electrically from these two power houses. The Canadian Niagara Power Company are erecting a power house to contain eleven units of 10,250 h.p. each, the turbines being twin inward flow reaction turbines. The Electrical Development Com- pany of Ontario are erecting a power house to contain 1 1 units of 12,500 h.p. each. The Ontario Power Company are carrying out another scheme for developing 200,000 h.p. by twin inward flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and Manufacturing Company on the United States side have a station giving 35,000 h.p. and are constructing another to furnish 100,000 h.p. The mean flow of the Niagara river is about 222,000 cub. ft. per second with a fall of 1 60 ft. The works in progress if completed will utilize 650,000 h.p. and require 48,000 cub. ft. per second or 21 J% of the mean flow of the river (Unwin, " The Niagara Falls Power Stations," Proc. Inst. Mech. Eng., 1906). § 189. Different Forms of Turbine Wheel. — The wheel of a turbine or part of the machine on which the water acts is an annular space, furnished with curved vanes dividing it into passages exactly or roughly rectangular in cross section. For radial flow turbines the wheel may have the form A or B, fig. 192, A being most usual with B M »", FIG. 192. TURBINES] HYDRAULICS 101 FIG. 193. ) ) ' inward, and B with outward flow turbines. In A the wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallej flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders. Theory of Reaction Turbines. § 190. Velocity of Whirl and Velocity of Flow. — Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity », in the direction of a tangent to the path. That velocity may be resolved into two components, a whirl- ing velocity w in the direction of the wheel's rotation at the point c, and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial now turbines. This second component is termed the velocity of flow. Let Dp, wa, ua be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and ti,-, Wi, ti the same quantities at the inlet surface of the wheel. Let a and /3 be the angles which the water's direction of motion makes with the direction of motion of the wheel at those surfaces. Then Wo=v0 cos/3; ua=v, Wi=ViCOsa; «,=!)< sin a The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let ft, ft be the areas of the outlet and inlet surfaces of the wheel, and Q the volume of water passing through the wheel per second ; then »o = Q/ft.; w = Q/ft. (u) Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes), Q<, = 2irrada; ft =2«- be the angles the wheel vanes make with the inlet and outlet surfaces ; then cos COS , , FIG. 195 equations which may be used to determine and 0. § 192. Condition determining the Angle of the Vanes at the Outlet Surface of the Wheel. — It has been shown that, when the water leaves the wheel, it should ^ r^x~ have no tangential -^"^ velocity, if the effici- ency is to be as great as possible ; that is, w0 = o. Hence, from (10), cos /3 = o, 0 = 90°, Up=v,, and the direction of the water's motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines. Drawing v, or u, radial or axial as the case may be, and V0 tangential to the direction of motion, v,, can be found by the parallelogram of velocities. From fig. 195, tan <#> = i>«/V» = «o/V<>; (14) but is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when the velocity of flow «<> and velocity of the wheel V0 are known. When is thus determined, Correction of the Angle to allow for Thickness of Vanes. — In determining , it is most convenient to calculate its value approxi- mately at first, from a value of u, obtained by neglecting the thick- ness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to allow for the vane thickness. Let <*>' = tan-' («„ /V0) = tan-1 (Q/ft V0) be the first or approximate value of , and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle ', their width measured on that surface is t cosec '. Hence the space occupied by the vanes on the outlet surface is For A, fig. 192, ntdo cosec T B, fig. 192, ntd cosec f" (15) C, fig. 192, ntfa-n) cosec ) Call this area occupied by the vanes u. Then the true value of the clear discharging outlet of the wheel is ft — w, and the true value of ua is Q/(ft - w). The corrected value of the angle of the vanes will be B2/V.(0,-i»)J. (16) § 193. Head producing Velocity with which the Water enters the Wheel. — Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a hori- zontal plane, there is no work done by gravity on the water passing through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account. 102 HYDRAULICS [TURBINES Let V,-, V, be the velocities of the wheel at the inlet and outlet surfaces, Vi, v0 the velocities of the water, «,-, MO the velocities of flow, tVi.tVo the relative velocities, hi, h, the pressures, measured in feet of water, Ti, Ta the radii of the wheel, a the angular velocity of the wheel. At any point in the path of a portion of water, at radius r, the velocity v of the water may be resolved into a component V = or equal to the velocity at that point of the wheel, and a relative com- ponent »r. Hence the motion of the water may be considered to consist of two parts: — (a) a motion identical with that in a forced vortex of constant angular velocity o; (6) a flow along curves parallel to the wheel vane curves. Taking the latter first, 'and using Bernoulli's theorem, the change of pressure due to flow through the wheel passages is given by the equation h'i+Vri*/2g=h'.+Vn>/2g; A'i-A'. = (tV,,'-«ViJ)/2g. The variation of pressure due to rotation in a forced vortex is A'i-A'. = (Vi4-V.s)/2g. Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel is hi-ho^h'i+h'i-h'.-h'. = (Vi2- V.')/2g + (^'-IV , and of this hi—h, is expended in overcoming the pressure in the wheel, the velocity of flow into the wheel is J'i=<;.V(2g(H-f)-(Vi2-V,,'/2g + (lV<,1-tVi')/2g)|, (18) where c, may be taken 0-96. From (140), It will be shown immediately that or, as this is only a small term, and 9 is on the average 90°, we may take, for the present purpose, v,i =«< nearly. Inserting these values, and remembering that for an axial flow turbine V< = V,,, Ij =o, and the fall d in the wheel is to be added, For an outward flow turbine, For an inward flow turbine, § 194. Angle which the Guide-Blades make with the Circumference of the Wheel. — At the moment the water enters the wheel, the radial component of the velocity is tt,-, and the velocity is Vi. Hence, if 7 is the angle between the guide-blades and a tangent to the wheel •X = sin-'(«tM). This angle can, if necessary, be corrected to allow for the thickness of the guide-blades. § 195. Condition determining the Angle of the Vanes at the Inlet Surface of the Wheel. — The single condition necessary to be satisfied at the inlet surface of the wheel is that the *•«»_ water should enter the wheel without shock. This condition is satis- fied if the direction of relative motion of the water and wheel is parallel to the first element of the wheel vanes. Let A (fig. 196) be a point on the inlet sur- face of the wheel, and let Vi represent in magnitude and direc- tion the velocity of the water entering the wheel, and V,- the velocity of the wheel. Completing the parallelogram, t)ri is the direction of relative motion. Hence the angle between iv< and V, is the angle 6 which the vanes should make with the inlet surface of the wheel. § 196. Example of the Method of designing a Turbine. Professor James Thomson's Inward Flow Turbine. — Let H =the available fall after deducting loss of head in pipes and channels from the gross fall ; Q = the supply of water in cubic feet per second; and ij =the efficiency of the turbine. The work done per second is »;GQH, and the horse-power of the turbine is h.p. =))GQH/55O. If i\ is taken at 0-75, an allowance will be made for the frictional losses in the turbine, the leakage and the friction of the turbine shaft. Then h.p. = o-o8sQH. The velocity of flow through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be taken tti=«<,=0-I25V2gH, in which case about jfoth of the energy of the fall is carried away by the water discharged. The areas of the outlet and inlet surface of the wheel are then 27rr,,(f0 = 2xri(/i=Q/o-i25V (2gH). If we take r0, so that the axial velocity of discharge from the central orifices of the wheel is equal to «„, we get da = ra. If, to obtain considerable steadying action of the centrifugal head, Ti =2re, then di = %da. Speed of the Wheel. — Let Vi = o-66V2gH, or the speed due to half the fall nearly. Then the number of rotations of the turbine per second is N = Vi/2iri = i -Q579V (HV H/Q) ; also V«=Vir0/ri=o-33V2gH. Angle of Vanes with Outlet Surface. Tan = tto/Vo = 0-125/0-33 = -3788 ; (jf> = 21 "nearly. If this value is revised for the vane thickness it will ordinarily become about 25°. Velocity with which the Water enters the Wheel. — The head pro- ducing the velocity is H - (Vi'/2g) (i +«oWi2) +tti"/2g = H|i --4356(1 +0-0358) + -0156) Then the velocity is Vi = -96V2g(-5646H) =0-721 V~2gH". Angle of Guide-Blades. Siny = Ui/Vi =0-125/0-721 =0-173; 7 = 10° nearly. Tangential Velocity of Water entering Wheel. 0-7101 V 2gH. 1 25 = -4008; Angle of Vanes at Inlet Surface. Cot 8 = (wi— Vi)/tti = (- 6 = 68 "nearly. Hydraulic Efficiency of Wheel. = °-9373- This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be 0-75 to 0-80. Impulse and Partial Admission Turbines. § 197. The principal defect of most turbines with complete admission is the imperfection of the arrangements for working with less than the normal supply. With many forms of reaction turbine the efficiency is considerably reduced when the regulating TURBINES] HYDRAULICS 103 sluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest possible amount of work. The imperfection of the regulating arrangements is therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. B. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial flow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and guide-blades. Fontaine similarly employed two concentric axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but they are not applicable to every form of turbine. C. Gallon, in 1840, patented an arrangement of sluices for axial or outward flow turbines, which were to be closed success- ively as the water supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The water was thus admitted to opposite but equal arcs of the wheel, and the forces driving the turbine were sym- metrically placed. As soon as this arrangement was adopted, FIG. 197. a modification of the mode of action of the water in the turbine became necessary. If the turbine wheel passages remain full of water during the whole rotation, the water contained in each passage must be put into motion each time it passes an open portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must be placed above the tail water, and the wheel passages be allowed to clear themselves of water, while passing from one open portion of the sluices to the next. But if the wheel passages are free of water when they arrive at the open guide passages, then there can be no pressure other than atmospheric pressure in the clearance space between guides and wheel. The water must issue from the sluices with the whole velocity due to the head.; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a small final velocity only, precisely in the same way as when a single jet strikes a curved vane in the free air. Turbines of this kind are therefore termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action on the wheel, and the whole energy of the jet is im- parted to the wheel, simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efficiency of the wheel. The diameter of the wheel may be made as large as convenient, and the water admitted to a small fraction of the circumference only. Then the number of revolutions is independent of the water velocity, and may be kept down to a manageable value. § 198. General Description of an Impulse Turbine or Turbine with Free Deviation. — Fig. 197 shows a general sectional elevation of a Girard turbine, in n n n n which the flow is axial. The water, a a la L admitted above a horizontal floor, passes down through the annular wheel containing the guide- blades G, G, and thence into the re- volving wheel WW. The revolving wheel is fixed to a hollow shaft suspended from the pivot p. The solid internal shaft ii is merely a fixed column supporting the pivot. The advantage of this ,-. is that the pivot is should, however, be corrected for the space occupied by the guide-blades. The tangential velocity of the entering water is TVi =Vi COS-y=0-82V2g(H-b). The circumferential velocity of the wheel may be (at mean radius) Vi = o-sV2g(H-b). Hence the vane angle at inlet surface is given by the equation cot0=(«ii-Vi)/tti = (o-82-o-5)/o-45 = -7i; 0 = 55°- The relative velocity of the water striking the vane at the inlet edge is iv>=tt; cosec0 = \-22Ui. This relative velocity remains unchanged during the passage of the water over the vane; conse- quently the relative velocity at the point of discharge is vra = I-22M,. Also in an axial flow turbine V<, = Vi. If the final velocity of the water is axial, then cos^ = V0/»r<, = Vi/t)ri=o-5/(i-22Xo-45)=cos24° 23'. This should be corrected for the vane thickness. Neglecting this, U0 = vrosm =vrisin <#> = M,- cosec 0 sin <£ = o-5«;. The discharging area of the wheel must therefore be greater than the inlet area in the ratio of at least 2 to I. In some actual turbines the ratio is 7 to 3. This greater outlet area is obtained by splaying the wheel, as shown in the section (fig. 199). § 200. Pelton Wheel. — In the mining district of California about 1860 simple impulse wheels were used, termed hurdy-gurdy wheels. The wheels rotated in a vertical plane, being supported on a hori- zontal axis. Round the circumference were fixed flat vanes which were struck normally by a jet from a nozzle of size varying with the head and quantity of water. Such wheels have in fact long been used. They are not efficient, but they are very simply constructed. Then attempts were made to improve the efficiency, first by using hemispherical cup vanes, and then by using a double cup vane with a central dividing ridge, an arrangement invented by Pelton. In this last form the water from the nozzle passes half to each side of the wheel, just escaping clear of the backs of the advancing buckets. Fig. 203 shows a Pelton vane. Some small modifications have been made FIG. 203. by other makers, but they are not of any great importance. Fig. 204 shows a complete Pelton wheel with frame and casing, supply pipe and nozzle. Pelton wheels have been very largely used in America and to some extent in Europe. They are extremely simple and easy to construct or repair and on falls of 100 ft. or more are very efficient. The jet strikes tangentially to the mean radius of the buckets, and the face of the buckets is not quite radial but at right angles to the direction of the jet at the point of first impact. For greatest efficiency the peripheral velocity of the wheel at the mean radius of the buckets should be a little less than half the velocity of the jet. As the radius of the wheel can be taken arbitrarily, the number of revolutions per minute can be accommodated to that of the machinery to be driven. Pelton wheels have been made as small \ ,-ir\\. !$? I i "\« •-./ .- •:- •<: \ /' /A> i ; FIG. 204. as 4 in. diameter, for driving sewing machines, and as large as 24 ft. The efficiency on high falls is about 80 %. When large power is required two or three nozzles are used delivering on one wheel. The width of the buckets should be not less than seven times the diameter of the jet. At the Comstock mines, Nevada, there is a 3&-in. Pelton wheel made of a solid steel disk with phosphor bronze buckets riveted to the rim. The head is 2100 ft. and the wheel makes 1 150 revolutions per minute, the peripheral velocity being 180 ft. per sec. With a \-\n. nozzle the wheel uses 32 cub. ft. of water per minute and develops 100 h.p. At the Chollarshaft, Nevada, there are six Pelton wheeli on a fall of 1680 ft. driving electrical generators. With f-in. nozzles each develops 125 h.p. § 201. Theory of the Pelton Wheel.— Suppose a jet with a velocity » strikes tangentially a curved vane AB (fig. 205) moving in the same direction with the velocity u. The water will flow over the vane with the relative velocity » — u and at B will have the tangential TURBINES] HYDRAULICS relative velocity » — u making an angle a with the direction of the vane's motion. Combining this with the velocity u of the vane, the absolute velocity of the water leaving the vane will bew = Be. The com- ponent of w in the direction of motion of the vane is Bo = B6 — ab = u — (v — u) cos a. Hence if Q is the quantity of water reaching the vane per second the change of momentum per second in the direction of the vane's motion is (GQ/g)[f — {u — (r-tt)coso)] = (GQ/g)(v-u) (l + cos a). If a = 0°, cos 0 = 1, and the change of momentum per second, which is equal to the effort driving the vane, is P = 2(GQ/g) (»-«)• The work done on the vane is P«-3(GQ/f)(p-«)«. If a series of vanes are inter- posed in succession, the quantity of water imping- ing on the vanes per second is the total discharge of the nozzle, and the energy expended at the nozzle is GQi^g. Hence the efficiency of the arrangement is, when a = o°, neglecting friction, FIG. 205. which is a maximum and equal to unity if u = \v. In that case the whole energy of the jet is usefully expended in driving the series of vanes. In practice a cannot be quite zero or the water leaving one vane would strike the back of the next advancing vane. Fig. 203 shows a Pelton vane. The water divides each way, and leaves the vane on each side in a direction nearly parallel to the direction of motion of the vane. The best velocity of the vane is very approxi- mately half the velocity of the jet. § 202. Regulation of the Pelton Wheel. — At first Pelton wheels were adjusted to varying loads merely by throttling the supply. This method involves a total loss of part of the head at the sluice or throttle valve. In addition as the working head is reduced, the relation between wheel velocity and jet velocity is no longer that of greatest efficiency. Next a plan was adopted of deflecting the jet so that only part of the water reached the wheel when the Toad was reduced, the rest going to waste. This involved the use of an equal quantity of water for large and small loads, but it had, what in some cases is an advantage, the effect of preventing any water hammer in the supply pipe due to the action of the regulator. In most cases now regulation is effected by varying the section of the jet. A conical needle in the nozzle can be advanced or withdrawn so as to occupy more or less of the aperture of the nozzle. Such a needle can be controlled by an ordinary governor. § 203. General Considerations on the Choice of a Type of Turbine. — -The circumferential speed of any turbine is necessarily a fraction of the initial velocity of the water, and therefore is greater as the head is greater. In reaction turbines with com- plete admission the number of revolutions per minute becomes inconveniently great, for the diameter cannot be increased beyond certain limits without greatly reducing the efficiency. In impulse turbines with partial admission the diameter can be chosen arbitrarily and the number of revolutions kept down on high falls to any desired amount. Hence broadly reaction turbines are better and less costly on low falls, and impulse turbines on high falls. For variable water flow impulse turbines have some advantage, being more efficiently regulated. On the other hand, impulse turbines lose efficiency seriously if their speed varies from the normal speed due to the head. If the head is very variable, as it often is on low falls, and the turbine must run at the same speed whatever the head, the impulse turbine is not suitable. Reaction turbines can be constructed so as to overcome this difficulty to a great extent. Axial flow turbines with vertical shafts have the disadvantage that in addition to the weight of the turbine there is an unbalanced water pressure to be carried by the footstep or collar bearing. In radial flow turbines the hydraulic pressures are balanced. The application of turbines to drive dynamos directly has involved some new con- ditions. The electrical engineer generally desires a high speed of rotation, and a very constant speed at all times. The reaction turbine is generally more suitable than the impulse turbine. As the diameter of the turbine depends on the quantity of water and cannot be much varied without great inefficiency, a difficulty arises on low falls. This has been met by constructing four independent reaction turbines on the same shaft, each having of course the diameter suitable for one-quarter of the whole dis- charge, and having a higher speed of rotation than a larger turbine. The turbines at Rheinfelden and Chevres are so con- structed. To ensure constant speed of rotation when the head varies considerably without serious inefficiency, an axial flow turbine is generally used. It is constructed of three or four concentric rings of vanes, with independent regulating sluices, forming practically independent turbines of different radii. Any one of these or any combination can be used according to the state of the water. With a high fall the turbine of largest radius only is used, and the speed of rotation is less than with a turbine of smaller radius. On the other hand, as the fall decreases the inner turbines are used either singly or together, according to the power required. At the Zurich waterworks there are turbines of 90 h.p. on a fall varying from ic4 ft. to 4! ft. The power and speed are kept constant. Each turbine has three concentric rings. The outermost ring gives 90 h.p. with 105 cub. ft. per second and the maximum fall. The outer and middle compartments give the same power with 140 cub. ft. per second and a fall of 7 ft. 10 in. All three compartments working together develop the power with about 250 cub. ft. per second. In some tests the efficiency was 74% with the outer ring working alone, 75-4% with the outer and middle ring working and a fall of 7 ft., and 80-7% with all the rings working. § 204. Speed Governing. — When turbines are used to drive dynamos direct, the question of speed regulation is of great im- portance. Steam engines using a light elastic fluid can be easily regulated by governors acting on throttle or expansion valves. It is different with water turbines using a fluid of great inertia. IV Hand Regulator FIG. 206. In one of the Niagara penstocks there are 400 tons of water flowing at 10 ft. per second, opposing enormous resistance to rapid change of speed of flow. The sluices of water turbines also are necessarily large and heavy. Hence relay governors must be xiv. 4 a io6 HYDRAULICS [PUMPS used, and the tendency of relay governors to hunt must be overcome. In the Niagara Falls Power House No. i, each tur- bine has a very sensitive centrifugal governor acting on a ratchet relay. The governor puts into gear one or other of two ratchets driven by the turbine itself. According as one or the other ratchet is in gear the sluices are raised or lowered. By a sub- sidiary arrangement the ratchets are gradually put out of gear unless the governor puts them in gear again, and this prevents the over correction of the speed from the lag in the action of the governor. In the Niagara Power House No. 2, the relay is an hydraulic relay similar in principle, but rather more complicated in arrangement, to that shown in fig. 206, which is a governor used for the 1250 h.p. turbines at Lyons. The sensitive governor G opens a valve and puts into action a plunger driven by oil pressure from an oil reservoir. As the plunger moves forward it gradually closes the oil admission valve by lowering the fulcrum end/ of the valve lever which rests on a wedge w attached to the plunger. If the speed is still too high, the governor re- opens the valve. In the case of the Niagara turbines the oil pressure is 1200 Ib per sq. in. One millimetre of movement of the governor sleeve completely opens the relay valve, and the relay plunger exerts a force of 50 tons. The sluices can be completely opened or shut in twelve seconds. The ordinary variation of speed of the turbine with varying load does not exceed i%. If all the load is thrown off, the momentary variation of speed is not more than 5 %. To prevent hydraulic shock in the supply pipes, a relief valve is provided which opens if the pressure is in excess of that due to the head. § 205. The Hydraulic Ram. — The hydraulic ram is an arrange- ment by which a quantity of water falling a distance h forces a portion of the water to rise to a height hi, greater than //. It consists of a supply reservoir (A, fig. 207), into which the water enters from some natural stream. A pipe s of considerable length conducts the water to a lower level, where it is discharged intermittently through a self-acting pulsating valve at d. The supply pipe s may be fitted with a flap valve for stopping the ram, and this is attached in some cases to a float, so that the ram starts and stops itself automatically, according as the supply cistern fills or empties. The lower float is just sufficient to keep open the flap after it has been raised by the action of the upper float. The length of chain is adjusted so that the upper float opens the flap when the level in the cistern is at the desired height. If the water-level falls below the lower float the flap closes. The pipe i should be as long and as straight as possible, and as it is subjected to considerable pressure from the sudden arrest of the motion of the water, it must be strong and strongly FIG. 208. FIG. 207. jointed, a is an air vessel, and e the delivery pipe leading to the reservoir at a higher level than A, into which water is to be pumped. Fig. 208 shows in section the construction of the ram itself, d is the pulsating discharge valve already mentioned, which opens inwards and downwards. The stroke of the valve is regulated by the cotter through the spindle, under which are washers by which the amount of fall can be regulated. At o is a delivery valve, opening outwards, which is often a ball- valve but sometimes a flap-valve. The water which is pumped passes through this valve into the air vessel a, from which it flows by the delivery pipe in a regular stream into the cistern to which the water is to be raised. In the vertical chamber behind the outer valve a small air vessel is formed, and into this opens an aperture J in. in diameter, made in a brass screw plug b. The hole is reduced to -jV in. in diameter at the outer end of the plug and is closed by a small valve opening inwards. Through this, during the rebound after each stroke of the ram, a small quantity of air is sucked in which keeps the air vessel supplied with its elastic cushion of air. During the recoil after a sudden closing of the valve d, the pressure below it is diminished and the valve opens, permitting outflow. In consequence of the flow through this valve, the water in the supply pipe acquires a gradually increasing velocity. The upward flow of the water, towards the valve d, increases the pressure tending to lift the valve, and at last, if the valve is not too heavy, lifts and closes it. The forward mo- mentum of the column in the supply pipe being destroyed by the stoppage of the flow, the water exerts a pressure at the end of the pipe sufficient to open the delivery valve o, and to cause a portion of the water to flow into the air vessel. As the water in the supply pipe comes to rest and recoils, the valve d opens again and the operation is repeated. Part of the energy of the descending column is employed in compressing the air at the end of the supply pipe and expanding the pipe itself. This causes a recoil of the water which momentarily diminishes the pressure in the pipe below the pressure due to the statical head. This assists in opening the valve d. The recoil of the water is sufficiently great to enable a pump to be attached to the ram body instead of the direct rising pipe. With this arrangement a ram working with muddy water may be employed to raise clear spring water. Instead of lifting the delivery valve as in the ordinary ram, the momentum of the column drives a sliding or elastic piston, and the recoil brings it back. This piston lifts and forces alternately the clear water through ordinary pump valves. PUMPS § 206. The different classes of pumps corre- spond almost exactly to the different classes of water motors, although the mechanical details of the construction are somewhat different. They are properly reversed water motors. Ordinary reciprocating pumps corre- spond to water-pressure engines. Chain and bucket pumps are in principle similar to water wheels in which the water acts by weight. Scoop wheels are similar to undershot water wheels, and centrifugal pumps to turbines. Reciprocating Pumps are single or double acting, and differ from water-pressure engines in that the valves are moved by the water instead of by automatic machinery. They may be classed thus: — 1. Lift Pumps. — The water drawn through a foot valve on the ascent of the pump bucket is forced through the bucket valve when it descends, and lifted by the bucket when it reascends. Such pumps give an intermittent discharge. 2. Plunger or Force Pumps, in which the water drawn through the foot valve is displaced by the descent of a solid plunger, and forced through a delivery valve. They have the advantage that PUMPS] HYDRAULICS 107 the friction is less than that of lift pumps, and the packing round the plunger is easily accessible, whilst that round a lift pump bucket is not. The flow is intermittent. 3. The Double-acting Force Pump is in principle a double plunger pump. The discharge fluctuates from zero to a maximum and back to zero each stroke, but is not arrested for any appreciable time. 4. Bucket and Plunger Pumps consist of a lift pump bucket combined with a plunger of half its area. The flow varies as in a double-acting pump. 5. Diaphragm Pumps have been used, in which the solid plunger is replaced by an elastic diaphragm, alternately depressed into and raised out of a cylinder. As single-acting pumps give an intermittent discharge three are generally used on cranks at 120°. But with all pumps the variation of velocity of discharge would cause great waste of work in the delivery pipes when they are long, and even danger from the hydraulic ramming action of the long column of water. An air vessel is interposed between the pump and the delivery pipes, of a volume from 5 to 100 times the space described by the plunger per stroke. The air in this must be replenished from time to time, or continuously, by a special air-pump. At low speeds not exceeding 30 ft. per minute the delivery of a pump is about 90 to 95% of the volume described by the plunger or bucket, from 5 to 10% of the discharge being lost by leakage. At high speeds the quantity pumped occasionally exceeds the volume described by the plunger, the momentum of the water keeping the valves open after the turn of the stroke. The velocity of large mining pumps is about 140 ft. per minute, the indoor or suction stroke being sometimes made at 250 ft. per minute. Rotative pumping engines of large size have a plunger speed of 90 ft. per minute. Small rotative pumps are run faster, but at some loss of efficiency. Fire-engine pumps have a speed of 1 80 to 220 ft. per minute. The efficiency of reciprocating pumps varies very greatly. Small reciprocating pumps, with metal valves on lifts of 15 ft., were found by Morin to have an efficiency of 16 to 40%, or on the average 25%. When used to pump water at considerable pressure, through hose pipes, the efficiency rose to from 28 to 57%, or on the average, with 50 to 100 ft. of lift, about 50%. A large pump with barrels 18 in. diameter, at speeds under 60 ft. per minute, gave the following results: — • Lift in feet . . . 14} 34 47 Efficiency .... -46 -66 -70 The very large steam-pumps employed for waterworks, with 150 ft. or more of lift, appear to reach an efficiency of 90%, not including the friction of the discharge pipes. Reckoned on the indicated work of the steam-engine the efficiency may be 80%. Many small pumps are now driven electrically and are usually three-throw single-acting pumps driven from the electric motor by gearing. It is not convenient to vary the speed of the motor to accommodate it to the]varying rate of pumping usually required. Messrs Hayward Tyler have introduced a mechanism for varying the stroke of the pumps (Sinclair's patent) from full stroke to nil, without stopping the pumps. § 207. Centrifugal Pump. — For large volumes of water on lifts not exceeding about 60 ft. the most convenient pump is the centrifugal pump. Recent improvements have made it available also for very high lifts. It consists of a wheel or fan with curved vanes enclosed in an annular chamber. Water flows in at the centre and is discharged at the periphery. The fan may rotate in a vertical or horizontal plane and the water may enter on one or both sides of the fan. In the latter case there is no axial unbalanced pressure. The fan and its casing must be filled with water before it can start, so that if not drowned there must be a foot valve on the suction pipe. When no special attention needs to be paid to efficiency the water may have a velocity of 6 to 7 ft. in the suction and delivery pipes. The fan often has 6 to 12 vanes. For a double-inlet fan of diameter D, the diameter of the inlets is D/2. If Q is the discharge in cub. ft. per second D = about 0-6 VQ in average cases. The peripheral speed is a little greater than the velocity due to the lift. Ordinary centrifugal pumps will have an efficiency of 40 to 60%. The first pump of this kind which attracted notice was one exhibited by J. G. Appold in 1851, and the special features of his pump have been retained in the best pumps since constructed. Appold's pump raised continuously a volume of water equal to 1400 times its own capacity per minute. It had no valves, and it permitted the passage of solid bodies, such as walnuts and oranges, without obstruction to its working. Its efficiency was also found to be good. Fig. 209 shows the ordinary form of a centrifugal pump. The pump disk and vanes B are cast in one, usually of bronze, FIG. 209. and the disk is keyed on the driving shaft C. The casing A has a spirally enlarging discharge passage into the discharge pipe K. A cover L gives access to the pump. S is the suction pipe which opens into the pump disk on both sides at D. Fig. 210 shows a centrifugal pump differing from ordinary centrifugal pumps in one feature only. The water rises through a suction pipe S, which divides so as to enter the pump wheel W at the centre on each side. The pump disk or wheel is very similar to a turbine wheel. It is keyed on a shaft driven by a belt on a fast and loose pulley arrangement at P. The water rotating in the pump disk presses outwards, and if the speed is sufficient a continuous flow is maintained through the pump and into the discharge pipe D. The special feature in this pump is that the water, discharged by the pump disk with a whirling velocity of not inconsiderable magnitude, is allowed to continue rotation in a chamber somewhat larger than the pump. The use of this whirlpool chamber was first suggested by Professor James Thomson. It utilizes the energy due to the whirling velocity of the water which in most pumps is wasted in eddies in the discharge pipe. In the pump shown guide-blades are also added which have the direction of the stream lines in a free vortex. They do not therefore interfere with the action of the water when pumping the normal quantity, but only prevent irregular motion. At A is a plug by which the pump case is filled before starting. If the pump is above the water to be pumped, a foot valve is required to permit the pump to be filled. Sometimes instead of the foot valve a delivery valve is used, an air-pump or steam jet pump being employed to exhaust the air from the pump case. § 208. Design and Proportions of a Centrifugal Pump. — The design of the pump disk is very simple. Let ri, ra be the radii of the inlet and outlet surfaces of the pump disk, di, r i (5) Variation of Pressure in the Pump Disk. — Precisely as in the case of turbines, it can be shown that the variation of pressure between the inlet and outlet surfaces of the pump is *.-*i = (V,«- Vi«)/2g - (lV.'-»ri!)/22. Inserting the values of v,,, Vn in (4) and (5), we get for normal conditions of working h,-hi = WJ- t-*J cosecV/2g+ («i2+Vi2)/2g = Vo2/2g - Mo2 COSCC V/2g +«i2/2g. (6) Hydraulic Efficiency of the Pump. — Neglecting disk friction, journal friction, and leakage, the efficiency of the pump can be found in the same way as that of turbines (§ 186). Let M be the moment of the couple rotating the pump, and a its angular velocity; wot ra the tangential velocity of the water and radius at the outlet surface; wt, n the same quantities at the inlet surface. Q being the discharge per second, the change of angular momentum per second is (GQ/g)(w<,r0— win). Hence M = (GQ/g)(war0—Win). In normal working, wi = o. Also, multiplying by the angular velocity, the work done per second is Ma = ( But the useful efficiency is work done in pumping is GQH. Therefore the § 209. Case I. Centrifugal Pump with no Whirlpool Chamber. — When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity »„, and impinges on a mass flowing to the discharge pipe at the much slower velocity v,. The radial component of va is almost necessarily wasted. From the tangential component there is a gain of pressure (W? -V ,2)/2g - (w, -».)*/22 = V,(W0— »,)/£, which will be small, if v, is small compared with wa. Its greatest value, if v, = %wa, is \w£\2g, which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure in the pump disk then balances the lift and the head tt;2/2g necessary to give the initial velocity of flow in the eye of the wheel. Mi2/2g + H = V02/2g-W02 COSCC 24>/2g+Mi2/2g, H = V02/2g - uf cosec 2/2g ) (8) or Vo = V(2gH+tt<,2 cosec * . ! and the efficiency of the pump is, from (7), l-»H/Vrf»,-fH/fy(V.-«. cot «)), = (V02-M«2 cosec V)/(2V.(V.-*. cot *!. (9) For<#>=oo0, i7=(V02-M<,2)/2V02, which is necessarily less than J. That is, half the work expended in driving the pump is wasted. By recurving the vanes, a plan intro- duced by Appold, the efficiency is increased, because the velocity v0 of discharge from the pump is diminished. If is very small, cosec = cot ; and then ij = (V0+«. cosec 0)/2V», which may approach the value I, as tends towards o. Equation (8) shows that u, cosec cannot be greater than V0. Putting M0 = o'2sV (2gH) we get the following numerical values o! the efficiency and the circumferential velocity of the pump : — PUMPS] HYDRAULICS 109 30° 20° 10° 0-47 0-56 0-65 o-73 0-84 i -06 I-I2 1-24 1-75 cannot practically be made less than 20°; and, allowing for the f fictional losses neglected, the efficiency of a pump in which = 20° is found to be about -60. §210. Case 2. Pump with a Whirlpool Chamber, as in fig. 210. — Professor James Thomson first suggested that the energy of the water after leaving the pump disk might be utilized, if a space were left in which a Free vortex could be formed. In such a free vortex the velocity varies inversely as the radius. The gain of pressure in the vortex chamber is, putting r0, ra for the radii to the outlet surface of wheel and to outside of free vortex, if k = ra/rn. The lift is then, adding this to the lift in the last case, H = (Vo'-wc,2 cosec^+ivKl -£2))/2g. But v,?=V+«o2 cosecV; .-.H ={(2-k2)VS-2kV0u0 cot -kW cosec2<£j/2g. (10) Putting this in the expression for the efficiency, we find a con- siderable increase of efficiency. Thus with <#> = 9O0 and fc = i, 17 = 5 nearly, a small angle and k = j, i) = I nearly. With this arrangement of pump, therefore, the angle at the outer ends of the vanes is of comparatively little importance. A moderate angle of 30° or 40° may very well be adopted. The following numerical values of the velocity of the circumference of the pump have been obtained by taking & = j, and w0 = o-25v (2gH). 45° -842 3°° "911 11 20° 1-023 „ The quantity of water to be pumped by a centrifugal pump neces- sarily varies, and an adjustment for different quantities of water can- not easily be introduced. Hence it is that the average efficiency of pumps of this kind is in practice less than the efficiencies given above. The advantage of a vortex chamber is also generally neglected. The velocity in the supply and discharge pipes is also often made greater than is consistent with a high degree of efficiency. Velocities of 6 or 7 ft. per second in the discharge and suction pipes, when the lift is small, cause a very sensible waste of energy; 3 to 6 ft. would be much better. Centrifugal pumps of very large size have been constructed. Easton and Anderson made pumps for the North Sea canal in Holland to deliver each 670 tons of water per minute on a lift of 5 ft. The pump disks are 8 ft. diameter. J. and H. Gwynne constructed some pumps for draining the Ferrarese Marshes, which together deliver 2000 tons per minute. A pump made under Pro- fessor J. Thomson's direction for drainage works in Barbados had a pump disk 16 ft. in diameter and a whirlpool chamber 32 ft. in diameter. The efficiency of centrifugal pumps when delivering less or more than the normal quantity of water is discussed in a paper in the Proc. Inst. Civ. Eng. vol. 53. § 211. High Lift Centrifugal Pumps. — It has long been known that centrifugal pumps could be worked in series, each pump overcoming a part of the lift. This method has been perfected, and centrifugal pumps for very high lifts with great efficiency have been used by Sulzer and others. C. W. Darley {Proc. Inst. Civ. Eng., supplement to vol. 154, p. 156) has described some pumps of this new type driven by Parsons steam turbines for the water supply of Sydney, N.S.W. Each pump was designed to deliver i \ million gallons per twenty-four hours against a head of 240 ft. at 3300 revs, per minute. Three pumps in series give therefore a lift of 720 ft. The pump consists of a central double- sided impeller 12 in. diameter. The water entering at the bottom divides and enters the runner at each side through a bell-mouthed passage. The shaft is provided with ring and groove glands which on the suction side keep the air out and on the pressure side prevent leakage. Some water from the pressure side leaks through the glands, but beyond the first grooves it passesinto a pocket and is returned to the suction side of the pump. For the glands on the suction side water is supplied from a low- pressure service. No packing is used in the glands. During the trials no water was seen at the glands. The following are the results of tests made at Newcastle: — I. II. III. IV. Duration of test . . hours 2 1-54 1-2 1-55 Steam pressure Ib per sq. in. 57 57 84 55 Weight of steam per water h.p. hour Ib 27-93 30-67 28-83 27-89 Speed in revs, per min. 3300 3330 3710 334° Height of suction . . .ft. ii ii II ii Total lift ft. 762 744 917 756 Million galls, per day pumped — By Venturi meter 1-573 1-499 1-689 I-503 By orifice 1-623 I-5I3 I-723 1-555 Water h.p 252 235 326 239 In trial IV. the steam was superheated 95° F. From other trials under the same conditions as trial I. the Parsons turbine uses 15-6 Ib of steam per brake h.p. hour, so that the combined efficiency of turbine and pumps is about 56%, a remarkably good result. § 212. Air-Lift Pumps. — An interesting and simple method of pumping by compressed air, invented by Dr J. Pohle of Arizona, is likely to be very useful in certain cases. Suppose a rising main placed in a deep bore hole in which there is a considerable depth of water. Air compressed to a sufficient pressure is con- veyed by an air pipe and introduced at the lower end of the rising main. The air rising in the main diminishes the average density of the contents of the main, and their aggregate weight no longer balances the pres- sure at the lower end of the main due to its sub- mersion. An up- ward flow is set up, and if the air supply is suffi- cient the water in the rising main is lifted to any required height. The higher the lift above the level in the bore hole the deeper must be the point at which air is injected. Fig. 212 shows an air- lift pump con- structed for W. H. Maxwell at the Tunbridge Wells water- works. There is a two-stage steam air compressor, compressing air to | FIG. 212. from 90 to 100 Ib per sq. in. The bore hole is 350 ft. deep, lined with steel pipes 1 5 in. diameter for 200 ft. and with perforated pipes 135 in. diameter for the lower 150 ft. The rest level of the water is 96 ft. from the ground-level, and the level when pumping 32,000 gallons per hour is 1 20 ft. from the ground-level. The rising main is 7 in. diameter, and is carried nearly to the bottom of the bore hole and to 20 ft. above the ground-level. The air pipe is 2\ in. diameter. In a trial run 31,402 gallons per hour were raised 133 ft. above the level in the well. Trials of the efficiency of the system made at San Francisco with varying conditions will be found in a paper by E. A. Rix (Journ. Amer. Assoc. Eng. Soc. vol. 25, -STeel Tubes 15 Diam. Rising Main 7 Diam. Air Pipt Zi' Diam no HYDRAZINE 1 900) . Maxwell found the best results when the ratio of immersion to lift was 3 to i at the start and 2-2 to i at the end of the trial. In these conditions the efficiency was 37% calculated on the indicated h.p. of the steam-engine, and 46% calculated on the indicated work of the compressor. 2-7 volumes of free air were used to i of water lifted. The system is suitable for temporary purposes, especially as the quantity of water raised is much greater than could be pumped by any other system in a bore hole of a given size. It is useful for clearing a boring of sand and may be advantageously used permanently when a boring is in sand or gravel which cannot be kept out of the bore hole. The initial cost is small. § 213. Centrifugal Fans. — Centrifugal fans are constructed similarly to centrifugal pumps, and are used for compressing air to pressures not exceeding 10 to 15 in. of water-column. With this small variation of pressure the variation of volume and density of the air may be neglected without sensible error. The conditions of pressure and discharge for fans are gener- ally less accurately known than in the case of pumps, and the design of fans is generally somewhat crude. They seldom have whirlpool chambers, though a large expanding outlet is pro- vided in the case of the important Guibal fans used in mine ventilation. It is usual to reckon the difference of pressure at the inlet and outlet of a fan in inches of water-column. One inch of water- column =64-4 ft. of air at average atmospheric pressure = 5-2lb per sq. ft. Roughly the pressure-head produced in a fan without means of utilizing the kinetic energy of discharge would be ti*/2g ft. of air, or 0-00024 »2 in. of water, where v is the velocity of the tips of the fan blades in feet per second. If d is the diameter of the fan and / the width at the external circumference, then wdt is the discharge area of the fan disk. If Q is the discharge in cub. ft. per sec., u=Q/-rdt is the radial velocity of discharge which is numerically equal to the discharge per square foot of outlet in cubic feet per second. As both the losses in the fan and the work done are roughly proportional to u* in fans of the same type, and are also proportional to the gauge pressure p, then if the losses are to be a constant percentage of the work done u may be taken proportional to V p. In ordinary cases u = about 22V p. The width / of the fan is generally from 0-35 to o-45) 80- 0 AS . a X2 1 i C L J ^ f «.«• ^* 4 1 i 1. 1' £• t o- 0 0 If p is the pressure difference in the fan in inches of water, and N the revolutions of fan, »=T / the kinetic energy is 0-00125 Q1"2 foot-pounds per second. The efficiency of fans is reckoned in two ways. If B.H.P. is the effective horse-power applied at the fan shaft, then the efficiency reckoned on the work of compression is On the other hand, if the kinetic energy in the delivery pipe is taken as part of the useful work the efficiency is ... Although the theory above is a rough one it agrees sufficiently with experiment, with some merely numerical modifications. An extremely interesting experimental investigation of the action of centrifugal fans has been made by H. Heenan and W. Gilbert (Proc. Inst. Civ. Eng. vol. 123, p. 272). The fans delivered through an air trunk in which different resistances could be obtained by intro- ducing diaphragms with circular apertures of different sizes. Suppose a fan run at constant speed with different resistances and the com- pression pressure, discharge and brake horse-power measured. The results plot in such a diagram as is shown in fig. 213. The less the resistance to discharge, that is the larger the opening in the air trunk, the greater the quantity of air discharged at the given speed of the fan. On the other hand the compression pressure diminishes. The curve marked total gauge is the compression pressure +the velocity head in th« discharge pipe, both in inches of water. This curve falls, but not nearly so much as the compression curve, when the resist- ance in the air trunk is diminished. The brake horse-power increases as the resistance is diminished because the volume of discharge in- creases very much. The curve marked efficiency is the efficiency calculated on the work of compression only. It is zero for no dis- charge, and zero also when there is no resistance and all the energy given to the air is carried away as kinetic energy. There is a dis- charge for which this efficiency is a maximum; it is about half the discharge which there is when there is no resistance and the delivery pipe is full open. The conditions of speed and discharge correspond- ing to the greatest efficiency of compression are those ordinarily taken as the best normal conditions of working. The curve marked 2000 3OOO Discharge - CfT. ptr mln. Tip Speed . too -ft. joer arc. FIG. 213. total efficiency gives the efficiency calculated on the work of com- pression and kinetic energy of discharge. Messrs Gilbert and Heenan found the efficiencies of ordinary fans calculated on the compression to be 40 to 60% when working at about normal conditions. Taking some of Messrs Heenan and Gilbert's results for ordinary fans in normal conditions, they have been found to agree fairly with the following approximate rules. Let pc be the compression pressure and q the volume discharged per second per square foot of outlet area of fan. Then the total gauge pressure due to pressure of compression and velocity of discharge is approximately: p = pe-\-Q-ooo$ft. per sec. The discharge per square foot of outlet of fan is — 9 = 15 to i8Vp cub. ft. per sec. The total discharge is t = -35^, 7 to 56 dt-Jp d = 0-22 to 0-25 V ( d=0-20 to 0-22V( /V p) ft. ft. For These approximate equations, which are derived purely from experiment, do not differ greatly from those obtained by the rough theory given above. The theory helps to explain the reason for the form of the empirical results. (W. C. U.) HYDRAZINE (DIAMIDOGEN), N2H< or H2 N-NH2, a compound of hydrogen and nitrogen, first prepared by Th. Curtius in 1887 from diazo-acetic ester, N2CH-CO2C2H6. This ester, which is obtained by the action of potassium nitrate on the hydrochloride of amidoacetic ester, yields on hydrolysis with hot concentrated potassium hydroxide an acid, which Curtius regarded as. CaHjN6(CO2H)8, but which A. Hantzsch and O. Silberrad (Ber., 1900, 33, p. 58) showed to be C2H2N4(CQ2H)2, bisdiazo- acetic acid. On digestion of its warm aqueous solution with warm dilute sulphuric acid, hydrazine sulphate and oxalic acid are obtained. C. A. Lobry de Bruyn (Ber., 1895, 28, p. 3085) prepared free hydrazine by dissolving its hydrochloride in methyl alcohol and adding sodium methylate; sodium chloride was precipitated and the residual liquid afterwards fractionated under reduced pressure. It can also be prepared by reducing potassium dinitrososulphonate in ice cold water by means of sodium amalgam: — HYDRAZONE— HYDROCEPHALUS in P. J. Sohestalcov (/. Russ. Phys. Chem. Soc., 1905, 37, p. i) obtained hydrazine by oxidizing urea with sodium hypochlorite in the presence of benzaldehyde, which, by combining with the hydrazine, protected it from oxidation. F. Raschig (German Patent 198307, 1908) obtained good yields by oxidizing ammonia with sodium hypochlorite in solutions made viscous with glue. Free hydrazine is a colourless liquid which boils at 113-5° C., and solidifies about o° C. to colourless crystals; it is heavier than water, in which it dissolves with rise of temperature. It is rapidly oxidized on exposure, is a strong reducing agent, and reacts vigorously with the halogens. Under certain conditions it may be oxidized to azoimide (A. W. Browne and F. F. Shetterly, /. Amer. C.S., 1908, p. 53). By fractional distilla- tion of its aqueous solution hydrazine hydrate NzHj-HjO (or perhaps H2N-NH3OH), a strong base, is obtained, which precipitates the metals from solutions of copper and silver salts at ordinary temperatures. It dissociates completely in a vacuum at 143°, and when heated under atmospheric pressure to 183° it decomposes into ammonia and nitrogen (A. Scott, J. Chem. Soc., 1904, 85, p. 913). The sulphate NjHLj-HzSO^ crystallizes in tables which are slightly soluble in cold water and readily soluble in hot water; it is decomposed by heating above 250° C. with explosive evolution of gas and liberation of sulphur. By the addition of barium chloride to the sulphate, a solution of the hydrochloride is obtained, from which the crystallized salt may be obtained on evaporation. Many organic derivatives of hydrazine are known, the most important being phenylhydrazine, which was discovered by Emil Fischer in 1877. It can be best prepared by V. Meyer and Lecco's method (Ber., 1883, 16, p. 2976), which consists in reducing phenyl- diazonium chloride in concentrated hydrochloric acid solution with stannous chloride also dissolved in concentrated hydrochloric acid. Phenylhydrazine is liberated from the hydrochloride so obtained by adding sodium hydroxide, the solution being then extracted with ether, the ether distilled off, and the residual oil purified by distilla- tion under reduced pressure. Another method is due to E. Bam- berger. The diazonium chloride, by the addition of an alkaline sulphite, is converted into a diazosulphonate, which is then reduced by zinc dust and acetic acid to phenylhydrazine potassium sulphite. This salt is then hydrolysed by heating it with hydrochloric acid — C,HsN2CI + K2SO, = KC1 + C6H6N2-SO,K, C6H6N2-SO*K + 2H = C,H6-NH.NH-SO3K, Phenylhydrazine is a colourless oily liquid which turns brown on exposure. It boils at 241° C., and melts at 17-5° C. It is slightly soluble in water, and is strongly basic, forming well-defined salts with acids. For the detection of substances containing the carbonyl group (such for example as aldehydes and ketones) phenylhydrazine is a very important reagent, since it combines with them with elimination of water and the formation of well-defined hydrazones (see ALDEHYDES, KETONES and SUGARS). It is a strong reducing agent; it precipitates cuprous oxide when heated with Fehling's solution, nitrogen and benzene being formed at the same time— C,H6-NH-NH2.+ 2CuO = Cu2O + N2+H2O + C.He. By energetic re- duction of phenylhydrazine (e.g. by use of zinc dust and hydrochloric acid), ammonia and aniline are produced — CeHsNH-NHj + 2H = CeH6NH2 + NH3. It is a]so a most important synthetic reagent. It combines with aceto-acetic ester to form phenylmethylpyrazolone, from which antipyrine (q.v.) may be obtained. Indoles (q.v.) are formed by heating certain hydrazones with anhydrous zinc chloride ; while semicarbazides, pyrrols (q.v.) and many other types of organic compounds may be synthesized by the use of suitable phenylhydrazine derivatives. HYDRAZONE, in chemistry, a compound formed by the con- densation of a hydrazine with a carbonyl group (see ALDE- HYDES ; KETONES). HYDROCARBON, in chemistry, a compound of carbon and hydrogen. Many occur in nature in the free state: for example, natural gas, petroleum and paraffin are entirely composed of such bodies; other natural sources are india-rubber, turpentine and certain essential oils. They are also revealed by the spectro- scope in stars, comets and the sun. Of artificial productions the most fruitful and important is provided by the destructive or dry distillation of many organic substances; familiar examples are the distillation of coal, which yields ordinary lighting gas, composed of gaseous hydrocarbons, and also coal tar, which, on subsequent fractional distillations, yields many liquid and solid hydrocarbons, all of high industrial value. For details reference should be made to the articles wherein the above subjects are treated. From the chemical point of view the hydrocarbons are of fundamental importance, and, on account of their great number, and still greater number of derivatives, they are studied as a separate branch of the science, namely, organic chemistry. See CHEMISTRY for an account of their classification, &c. HYDROCELE (Gr. vSup, water, and wjXij, tumour), the medical term for any collection of fluid other than pus or blood in the neighbourhood of the testis or cord. The fluid is usually serous. Hydrocele may be congenital or arise in the middle-aged without apparent cause, but it is usually associated with chronic orchitis or with tertiary syphilitic enlargements. The hydrocele appears as a rounded, fluctuating translucent swelling in the scrotum, and when greatly distended causes a dragging pain. Palliative treatment consists in tapping aseptically and remov- ing the fluid, the patient afterwards wearing a suspender. The condition frequently recurs and necessitates radical treatment. Various substances may be injected; or the hydrocele is incised, the tunica partly removed and the cavity drained. HYDROCEPHALUS (Gr. vSup, water, and K€<£aXi), head), a term applied to disease of the brain which is attended with excessive effusion of fluid into its cavities. It exists in two forms — acute and chronic hydrocephalus. Acute hydro- cephalus is another name for tuberculous meningitis (see MENINGITIS). Chronic hydrocephalus, or " water on the brain," consists in an effusion of fluid into the lateral ventricles of the brain. It is not preceded by tuberculous deposit or acute inflammation, but depends upon congenital malformation or upon chronic inflammatory changes affecting the membranes. When the disease is congenital, its presence in the foetus is apt to be a source of difficulty in parturition. It is however more commonly developed in the first six months of life; but it occasionally arises in older children, or even in adults. The chief symptom is the gradual increase in size of the upper part of the head out of all proportion to the face or the rest of the body. Occurring at an age when as yet the bones of the skull have not become welded together, the enlargement may go on to an enormous extent, the spaces between the bones becoming more and more expanded. In a well-marked case the deformity is very striking; the upper part of the forehead projects abnormally, and the orbital plates of the frontal bone being inclined forwards give a downward tilt to the eyes, which have also peculiar rolling movements. The face is small, and this, with the enlarged head, gives a remarkable aged expression to the child. The body is ill-nourished, the bones are thin, the hair is scanty and fine and the teeth carious or absent. The average circumference of the adult head is 22 in., and in the normal child it is of course much less. In chronic hydro- cephalus the head of an infant three months old has measured 29 in.; and in the case of the man Cardinal, who died in Guy's Hospital, the head measured 33 in. In such cases the head cannot be supported by the neck, and the patient has to keep mostly in the recumbent posture. The expansibility of the skull prevents destructive pressure on the brain, yet this organ is materially affected by the presence of the fluid. The cerebral ventricles are distended, and the convolutions are flattened. Occasionally the fluid escapes into the cavity of the cranium, which it fills, pressing down the brain to the base of the skull. As a consequence, the functions of the brain are interfered with, and the mental condition is impaired. The child is dull, listless and irritable, and sometimes imbecile. The special senses become affected as the disease advances; sight is often lost, as is also hearing. Hydrocephalic children generally sink in a few years; nevertheless there have been instances of persons with this disease living to old age. There are, of course, grades of the affection, and children may present many of the symptoms of it in a slight degree, and yet recover, the head ceasing to expand, and becoming in due course firmly ossified. 112 HYDROCHARIDEAE Various methods of treatment have been employed, but the results are unsatisfactory. Compression of the head by bandages, and the administration of mercury with the view of promoting absorption of the fluid, are now little resorted to. Tapping the fluid from time to time through one of the spaces between the bones, drawing off a little, and thereafter employing gentle pressure, has been tried, but rarely with benefit. Attempts have also been made to establish a permanent drainage between the interior of the lateral ventricle and the sub-dural space, and between the lumbar region of the spine and the abdomen, but without satisfactory results. On the whole, the plan of treatment which aims at maintaining the patient's nutrition by appropriate food and tonics is the most rational and successful. (E. O.*) HYDROCHARIDEAE, in botany, a natural order of Mono- cotyledons, belonging to the series Helobieae. They are water- plants, represented in Britain by frog-bit (Hydrocharis Morsus- ranae) and water-soldier (Stratiotes aloides). The order contains about fifty species in fifteen genera, twelve of which occur in fresh water while three are marine: and includes both floating and submerged forms. Hydrocharis floats on the surface of still water, and has rosettes of kidney-shaped leaves, from among which spring the flower-stalks; stolons bearing new leaf- rosettes are sent out on all sides, the plant thus propagating itself in the same way as the strawberry. Slratiotes alcfides has a rosette of stiff sword- like leaves, which when the plant is in flower project above the Surface; it is also stoloniferous, the young rosettes sinking to the bottom at the beginning of winter and rising again to the surface in the spring. Vallisneria (eel-grass) contains two species, one native of tropical Asia, the other in- habiting the warmer parts of both hemi- spheres and reaching as far north as south Morsus-ranae — Europe. It grows in FlG. I. — Hydrocharis the mud at the bottom of fresh water, and the short stem bears a cluster of long, narrow grass-like leaves; new plants are formed at Frog-bit — male plant, half natural size. 1, Female flower, half natural size. 2, Stamens, enlarged. 3, Barren pistil of male flower, enlarged. 4, Pistil of female flower. 5, Fruit. 6, Fruit cut transversely. 7 Seed 8, 9, Floral diagrams of male and female the end °f flowers respectively. runners. Another type s. Rudimentary stamens. is represented by Elodea canadensis or water-thyme, which has been introduced into the British Isles from North America. It is a small, submerged plant with long, slender branching stems bearing whorls of narrow toothed leaves; the flowers appear at the surface when mature. Halophila, Enhalus and Thalassia are submerged maritime plants found on tropical coasts, mainly in the Indian and Pacific oceans; Halophila has an elongated stem rooting at the nodes; Enhalus a short, thick rhizome, clothed with black threads resembling horse-hair, the persistent hard-bast strands of the leaves; Thalassia has a creeping rooting stem with upright branches bearing crowded strap-shaped leaves in two rows. The flowers spring from, or are enclosed in, a spathe, and are unisexual and regular, with generally a calyx and corolla, each of three members; the stamens are in whorls of three, the inner whorls are often barren; the two to fifteen carpels form an inferior ovary containing generally numerous ovules on often large, produced, parietal placentas. The fruit is leathery or fleshy, opening irregularly. The seeds contain a large embryo and no endosperm. In Hydrocharis (fig. i), which is dioe- cious, the flowers are borne above the surface of the water, have con- spicuous white petals, contain honey and are pollinated by in- sects. Stratiotes has similar flowers which come above the surface only for pollination, becoming sub- merged again during ripening of the fruit. In Val- lisneria (fig. 2), which is also dioe- cious, the small male flowers are borne in large numbers in short- stalked spathes; the petals are minute and scale- like, and only two of the three stamens are fer- FIG. 2.— Vallisneria spiralis—Ee\ grass — tile; the flowers 9ya,rter, natural size- A' Female plant; B, , . , Male plant, become detached before opening and rise to the surface, where the sepals expand and form a float bearing the two projecting semi-erect stamens. The female flowers are solitary and are raised to the surface on a long, spiral stalk; the ovary bears three broad styles, on which some of the large, sticky Af^A pollen-grains from the floating male flowers get de- posited (fig. 3). After pollination the female flower becomes drawn below the surface by the spiral con- traction of the long stalk, and the fruit ripens near the bottom. Elodea has poly- gamous flowers FIG. 3. (that is, male, female and hermaphrodite), solitary, in slender, tubular spathes; the male flowers become detached and rise to the surface; the females are raised to the surface when mature, and receive the floating pollen from the male. The flowers of Halophila are submerged and apetalous. The order is a widely distributed one; the marine forms are tropical or subtropical, but the fresh-water genera occur also in the temperate zones. HYDROCHLORIC ACID— HYDROGEN HYDROCHLORIC ACID, also known in commerce as " spirits of salts " and " muriatic acid," a compound of hydrogen and chlorine. Its chemistry is discussed under CHLORINE, and 'its manufacture under ALKALI MANUFACTURE. HYDRODYNAMICS (Gr. vdwp, water, 8vva/us, strength), the branch of hydromechanics which discusses the motion of fluids (see HYDROMECHANICS). HYDROGEN [symbol H, atomic weight 1-008(0=16)], one of the chemical elements. Its name is derived from Gr. OSoip, water, and yevvativ, to produce, in allusion to the fact that water is produced when the gas burns in air. Hydrogen appears to have been recognized by Paracelsus in the i6th century; the combustibility of the gas was noticed by Turquet de Mayenne in the i7th century, whilst in 1700 N. Lemery showed that a mixture of hydrogen and air detonated on the application of a light. The first definite experiments concerning the nature of hydrogen were made in 1766 by H. Cavendish, who showed that it was formed when various metals were acted upon by dilute sulphuric or hydrochloric acids. Cavendish called it " in- flammable air," and for some time it was confused with other inflammable gases, all of which were supposed to contain the 'Same inflammable principle, " phlogiston," in combination with varying amounts of other substances. In 1781 Cavendish showed that water was the only substance produced when hydrogen was burned in air or oxygen, it having been thought previously to this date that other substances were formed during the reaction, A. L. Lavoisier making many experiments with the object of finding an acid among the products of combustion. Hydrogen is found in the free state in some volcanic gases, in fumaroles, in the carnallite of the Stassfurt potash mines (H. Precht, Bcr., 1886, 19, p. 2326), in some meteorites, in certain stars and nebulae, and also in the envelopes of the sun. In combination it is found as a constituent of water, of the gases from certain mineral springs, in many minerals, and in most animal and vegetable tissues. It may be prepared by the electro- lysis of acidulated water, by the decomposition of water by various metals or metallic hydrides, and by the action of many metals on acids or on bases. The alkali metals and alkaline earth metals decompose water at ordinary temperatures; magnesium begins to react above 70° C., and zinc at a dull red heat. The decomposition of steam by red hot iron has been studied by H. Sainte-Claire Deville (Comptes rendus, 1870, 70, p. 1105) and by H. Debray (ibid., 1879, 88, p. 1341), who found that at about 1500° C. a condition of equilibrium is reached. H. Moissan (Bull. soc. chim., 1902, 27, p. 1141) has shown that potassium hydride decomposes cold water, with evolution of hydrogen, KH-r-H2O = KOH+H2. Calcium hydride or hydrolite, prepared by passing hydrogen over heated calcium, decomposes water similarly, i gram giving i litre of gas; it has been proposed as a commercial source (Prats Aymerich, Abst. J.C.S., 1907, ii. p. 543), as has also aluminium turnings moistened with potassium cyanide and mercuric chloride, which decomposes water regularly at 70°, i gram giving 1-3 litres of gas (Mauricheau-Beaupre, Comptes rendus, 1908, 147, p. 310). Strontium hydride behaves similarly. In preparing the gas by the action of metals on acids, dilute sulphuric or hydrochloric acid is taken, and the metals commonly used are zinc or iron. So obtained, it contains many impurities, such as carbon dioxide, nitrogen, oxides of nitrogen, phosphoretted hydrogen, arseniuretted hydrogen, &c., the removal of which is a matter of great difficulty (see E. W. Morley, Amer. Chem. Journ., 1890, 12, p. 460). When prepared by the action of metals on bases, zinc or aluminium and caustic soda or caustic potash are used. Hydrogen may also be obtained by the action of zinc on ammonium salts (the nitrate excepted) (Lorin, Comptes rendus, 1865, 60, p. 745) and by heating the alkali formates or oxalates with caustic potash or soda, Na2C2O4+2NaOH = H2-r-2Na2CO3. Technically it is prepared by the action of superheated steam on incandescent coke (see F. Hembert and Henry, Comptes rendus, 1885, 101, p. 797; A. Naumann and C. Pistor, Ber., 1885, 18, p. 1647), or by the electrolysis of a dilute solution of caustic soda (C. Winssinger, Chem. Zeit., 1898, 22, p. 609; " Die Elektrizitats-Aktiengesell- schaft," Zeit. f. Elektrochem., 1901, 7, p. 857). In the latter method a 15 % solution of caustic soda is used, and the electrodes are made of iron; the cell is packed in a wooden box, surrounded with sand, so that the temperature is kept at about 70° C.; the solution is replenished, when necessary, with distilled water. The purity of the gas obtained is about 97 %• Pure hydrogen is a tasteless, colourless and odourless gas of specific gravity 0-06947 (air= i) (Lord Rayleigh, Proc. Roy. Soc., 1893, p. 319). It may be liquefied, the liquid boiling at -252-68° C. to -252-84°C., and it has also been solidified, the solid melting at -264° C. (J. Dewar, Comptes rendus, 1899, 129, p. 451; Chem. News, 1901, 84, p. 49; see also LIQUID GASES). The specific heat of gaseous hydrogen (at constant pressure) is 3.4041 (water=i), and the ratio of the specific heat at constant pressure to the specific heat at constant volume is 1-3852 (W. C. Rontgen, Fogg. Ann., 1873, 148, p. 580). On the spectrum see SPECTROSCOPY. Hydrogen is only very slightly soluble in water. It diffuses very rapidly through a porous membrane, and through some metals at a red heat (T. Graham, Proc. Roy. Soc., 1867, 15, p. 223; H. Sainte-Claire Deville and L. Troost, Comptes rendus, 1863, 56, p. 977). Palladium and some other metals are capable of absorbing large volumes of hydrogen (especially when the metal is used as a cathode in a water electrolysis apparatus). L. Troost and P. Hautefeuille (Ann. chim. phys., 1874, (5) 2, p. 279) considered that a palladium hydride of composition Pd2H was formed, but the investigations of C. Hoitsema (Zeit. phys. Chem., 1895, 17, p. i), from the standpoint of the phase rule, do not favour this view, Hoitsema being of the opinion that the occlusion of hydrogen by palladium is a process of continuous absorption. Hydrogen burns with a pale blue non-luminous flame, but will not support the combustion of ordinary combustibles. It forms a highly explosive mixture with air or oxygen, especially when in the proportion of two volumes of hydrogen to one volume of oxygen. H. B. Baker (Proc. Chem. Soc., 1902, 18, p. 40) has shown that perfectly dry hydrogen will not unite with perfectly dry oxygen. Hydrogen combines with fluorine, even at very low temperatures, with great violence; it also combines with carbon, at the temperature of the electric arc. The alkali metals when warmed in a current of hydrogen, at about 360° C., form hydrides of composition RH(R = Na, K, Rb, Cs), (H. Moissan, Bull. soc. chim., 1902, 27, p. 1141); calcium and strontium similarly form hydrides CaH2, SrH2 at a dull red heat (A. Guntz, Comptes rendus, 1901, 133, p. 1209). Hydrogen is a very powerful re- ducing agent; the gas occluded by palladium being very active in this respect, readily reducing ferric salts to ferrous salts, nitrates to nitrites and ammonia, chlorates to chlorides, &c. For determinations of the volume ratio with which hydrogen and oxygen combine, see J. B. Dumas, Ann. chim. phys., 1843 (3), 8, p. 189; O. Erdmann ^nd R. F. Marchand, ibid. p. 212; E. H. Keiser, Ber., 1887, 20, p. 2323; J. P. Cooke and T. W. Richards, Amer. Chem. Journ., 1888, 10, p. 191; Lord Rayleigh, Chem. News, 1889, 59, p. 147; E. W. Morley, Zeit. phys. Chem., 1890, 20, p. 417; and S. A. Leduc, Comptes rendus, 1899, 128, p. 1158. Hydrogen combines with oxygen to form two definite com- pounds, namely, water (q.v.), H2O, and hydrogen peroxide, H2O2, whilst the existence of a third oxide, ozonic acid, has been indicated. Hydrogen peroxide, H2O2, was discovered by L. J. Thenard in 1818 (Ann. chim. phys., 8, p. 306). It occurs in small quantities in the atmosphere. It may be prepared by passing a current of carbon dioxide through ice-cold water, to which small quantities of barium peroxide are added from time to time (F. Duprey, Comptes rendus, 1862, 55, p. 736; A. J. Balard, ibid., p. 758), BaO2+CO2-r-H2O = H2O2+BaCO3. E. Merck (Abst. J.C.S., 1907, ii., p. 859) showed that barium percarbonate, BaC04, is formed when the gas is in excess; this substance readily yields the peroxide with an acid. Or barium peroxide may be decom- posed by hydrochloric, hydrofluoric, sulphuric or silicofluoric acids (L. Crismer, Bull. soc. chim., 1891 (3), 6, p. 24; Hanriot, Comptes rendus, 1885, 100, pp. 56, 172), the peroxide being added HYDROGRAPHY— HYDROLYSIS in small quantities to a cold dilute solution of the acid. It is necessary that it should be as pure as possible since the commercial product usually contains traces of ferric, manganic and aluminium oxides, together with some silica. To purify the oxide, it is dissolved in dilute hydrochloric acid until the acid is neatly neutralized, the solution is cooled, filtered, and baryta water is added until a faint permanent white precipitate of hydrated barium peroxide appears; the solution is now filtered, and a concentrated solution of baryta water is added to the filtrate, when a crystalline precipitate of hydrated barium peroxide, BaO28-H2O, is thrown down. This is filtered off and well washed with water. The above methods give a dilute aqueous solution of hydrogen peroxide, which may be concentrated somewhat by evaporation over sulphuric acid in vacua. H. P. Talbot and H. R. Moody (Jour. Anal. Chem., 1892, 6, p. 650) prepared a more concentrated solution from the commercial product, by the addition of a 10% solution of alcohol and baryta water. The solution is filtered, and the barium precipitated by sulphuric acid. The alcohol is removed by distillation in vacua, and by further concentration in vacua a solution may be obtained which evolves 580 volumes of oxygen. R. Wolffenstein (Ber., 1894, 27, p. 2307) prepared practically anhydrous hydrogen peroxide (containing 99-1% H20j) by first removing all traces of dust, heavy metals and alkali from the commercial 3% solution. The solution is then concentrated in an open basis on the water- bath until it contains 48% HjOj. The liquid so obtained is extracted with ether and the ethereal solution distilled under diminished pressure, and finally purified by repeated distillations. W. Staedel (Zeit.f. angew. Chem., 1902, 15, p. 642) has described solid hydrogen peroxide, obtained by freezing concentrated solutions. Hydrogen peroxide is also found as a product in many chemical actions, being formed when carbon monoxide and cyanogen burn in air (H. B. Dixon); by passing air through solutions of strong bases in the presence of such metals as do not react with the bases to liberate hydrogen; by shaking zinc amalgam with alcoholic sulphuric acid and air (M. Traube, Ber., 1882, 15, p. 659) ; in the oxidation of zinc, lead and copper in presence of water, and in the electrolysis of sulphuric acid of such strength that it contains two molecules of water to one molecule of sulphuric acid (M. Berthelot, Camples rendus, 1878, 86, p. 71). The anhydrous hydrogen peroxide obtained by Wolfienstein boils at 84-8s°C. (68 mm.) ; its specific gravity is 1-4996 (1-5° C.). It is very explosive (W. Spring, Zeit. anorg. Chem., 1895, 8, p. 424). The explosion risk seems to be most marked in the preparations which have been extracted with ether previous to distillation, and J. W. Briihl (Ber., 1895, 28, p. 2847) is of opinion that a very unstable, more highly oxidized product is produced in small quantity in the process. The solid variety prepared by Staedel forms colourless, prismatic crystals which melt at -2° C. ; it is decomposed with explosive violence by platinum sponge, and traces of manganese dioxide. The dilute aqueous solution is very unstable, giving up oxygen readily, and decomposing with explosive violence at 100° C. An aqueous solution containing more than 1-5% hydrogen peroxide reacts slightly acid. To- wards lupetidin [oa' dimethyl piperidine, C6HjN(CH3)2] hydrogen peroxide acts as a dibasic acid (A. Marcuse and R. Wolffenstein, Ber., 1001, 34, p. 2430; see also G. Bredig, Zeit. Electrochem., 1901, 7, p. 622). Cryoscopic determinations of its molecular weight show that it is H2O2. [G. Carrara, Rend, della Accad. dei Lincei, 1892 (5), i, ii. p. 19; W. R. Orndorff and J. White, Amer. Chem. Journ., 1893, 15, p. 347.] Hydrogen peroxide behaves very frequently as a powerful oxidizing agent; thus lead sulphide is converted into lead sulphate in presence of a dilute aqueous solution of the peroxide, the hydroxides of the alkaline earth metals are converted into peroxides of the type MOy8H2O, titanium dioxide is converted into the trioxide, iodine is liberated from potassium iodide, and nitriles (in alkaline solution) are converted into acid-amides (B. Radziszewski,5er., 1884, 17, p. 355). In many cases it is found that hydrogen peroxide will only act as an oxidant when in the presence of a catalyst; for example, formic, glygollic, lactic, tartaric, malic, benzoic and other organic acids are readily oxidized in the presence of ferrous sulphate (H. J. H. Fenton, Jour. Chem. Soc., 1900, 77, p. 69), and sugars are readily oxidized in the presence of ferric chloride (O. Fischer and M. Busch, Ber., 1891, 24, p. 1871). It is sought to explain these oxidation processes by assuming that the hydrogen peroxide unites with the compound undergoing oxidation to form an addition compound, which subsequently decomposes (J. H. Kastle and A. S. Loevenhart, Amer. Chem. Journ., 1903, 29, pp. 397, 517). Hydrogen peroxide can also react as a reducing agent, thus silver oxide is reduced with a rapid evolution of oxygen. The course of this reaction can scarcely be considered as definitely settled; M. Berthelot considers that a higher oxide of silver is formed, whilst A. Baeyer and V. Villiger are of opinion that reduced silver is obtained [see Comptes rendus, 1901, 133, p. 555; Ann. Chim. Phys., 1897 (7), n, p. 217, and .Ber., 1901,34, p. 2769]. Potassium permanganate, in the presence of dilute sulphuric acid, is rapidly reduced by hydrogen peroxide, oxygen being given off, 2KMnO4-(- 3H2SO4-r-5H2O2 = K2SO4-|-2MnSO4-r-8H2O+5O2. Lead peroxide is reduced to the monoxide. Hypochlorous acid and its salts, together with the corresponding bromine and iodine compounds, liberate oxygen violently from hydrogen peroxide, giving hydro- chloric, hydrobromic and hydriodic acids (S. Tanatar, Ber., 1899, 32, p. 1013). On the constitution of hydrogen peroxide see C. F. Schonbein, Jour. prak. Chem., 1858-1868; M. Traube, Ber., 1882-1889; J. W. Briihl, Ber., 1895, 28, p. 2847; 1900, 33, p. 1709; S. Tanatar, Ber., 1903. 36, p. 1893. Hydrogen peroxide finds application as a bleaching agent, as an antiseptic, for the removal of the last traces of chlorine and sulphur dioxide employed in bleaching, and for various quantitative separa- tions in analytical chemistry (P. Jannasch, Ber., 1893, 26, p. 2908). It may be estimated by titration with potassium permanganate in acid solution; with potassium ferricyanide in alkaline solution, 2K,Fe(CN)8+2KOH+H2O2 = 2K4Fe(CN)«+2H2O+O2;or by oxidiz- ing arsenious acid in alkaline solution with the peroxide and back titration of the excess of arsenious acid with standard iodine (B. Grutzner, Arch, der Pharm., 1899, 237, p. 705). It may be recognized by the violet coloration it gives when added to a very dilute solution of potassium bichromate in the presence of hydro- chloric acid ; by the orange-red colour it gives with a solution of titanium dioxide in concentrated sulphuric acid; and by the pre- cipitate of Prussian blue formed when it is added to a solution containing ferric chloride and potassium ferricyanide. Ozonic Acid, H2O«. By the action of ozone on a 40% solution of potassium hydroxide, placed in a freezing mixture, an orange- brown substance is obtained, probably K2O4, which A. Baeyer and V. Villiger (Ber., 1902, 35, p. 3038) think is derived from ozonic acid, produced according to the reaction Oa+H2O = H2O«. HYDROGRAPHY (Gr. vSup, water, and ypafaiv, to write), the science dealing with all the waters of the earth's surface, including the description of their physical features and con- ditions; the preparation of charts and maps showing the position of lakes, rivers, seas and oceans, the contour of the sea-bottom, the position of shallows, deeps, reefs and the direction and volume of currents; a scientific description of the position, volume, configuration, motion and condition of all the waters of the earth. See also SURVEYING (Nautical) and OCEAN AND OCEANOGRAPHY. The Hydrographic Department of the British Admiralty, established in 1795, undertakes the making of charts for the admiralty, and is under the charge of the hydrographer to the admiralty (see CHART). HYDROLYSIS (Gr. vSup, water, \vttv, to loosen), in chemistry, a decomposition brought about by water after the manner shown in the equation R-X+H-OH = R-H+X-OH. Modern research has proved that such reactions are not occasioned by water acting as H2O, but really by its ions (hydrions and hydroxidions), for the velocity is proportional (in accordance with the law of chemical mass action) to the concentration of these ions. This fact explains the so-called " catalytic " action of acids and bases in decomposing such compounds as the esters. The term " saponification " (Lat. sapo, soap) has the same meaning, but it is more properly restricted to the hydrolysis of the fats, i.e. glyceryl esters of organic acids, into glycerin and a soap (see CHEMICAL ACTION). HYDROMECHANICS HYDROMECHANICS (Gr. vdpo/jurixanKa) , the science of the mechanics of water and fluids in general, including hydrostatics or the mathematical theory of fluids in equilibrium, and hydro- mechanics, the theory of fluids in motion. The practical applica- tion of hydromechanics forms the province of hydraulics (g.v.) . Historical. — The fundamental principles of hydrostatics were first given by Archimedes in his work lUpi ran bxov^tv