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See also:DISCHARGE FROM ORIFICES] owners below the reservoirs a right to a regulated See also:supply through-out the See also:year. This See also:compensation See also:water requires to be measured in such a way that the millowners and others interested in the See also:matter can assure themselves that they are receiving a proper quantity, and they are generally allowed a certain amount of See also:control as to the times during which the daily supply is discharged into the stream. Fig. 74 shows an arrangement designed for the See also:Manchester water See also:works. The water enters from the See also:reservoir a chamber A, the See also:object of which is to still the irregular See also:motion of the water. The See also:admission is regulated by sluices at b, b, b. The water is discharged by orifices or notches at a, a, over which a tolerably See also:constant See also:head is maintained by adjusting the sluices at b, b, b. At any See also:time the millowners can see whether the discharge is given and whether the proper head is maintained over the orifices. To test at any time the discharge of the orifices, a gauging See also:basin B is provided. The water ordinarily55 flows over this, without entering it, on a See also:floor of See also:cast-See also:iron plates. If the discharge is to be tested, the water is turned for a definite time into the gauging basin, by suddenly opening and closing a sluice at c. The See also:volume of flow can be ascertained from the See also:depth in the gauging chamber. A See also:mechanical arrangement (fig. 73) was designed for securing an absolutely constant head over the orifices at a, a. The orifices were formed in a cast-iron See also:plate capable of sliding up and down, without sensible leakage, on the See also:face of the See also:wall of the chamber. The orifice plate was attached by a See also:link to a See also:lever, one end of which rested on the wall and the other on floats f in the chamber A. The floats See also:rose and See also:fell with the changes of level in the chamber, and raised and lowered the orifice plate at the same time. This mechanical arrangement was not finally adopted, careful watching of the sluices at b, b, b, being sufficient to secure a See also:regular discharge. The arrangement is then See also:equivalent to an See also:Italian See also:module, but on a large See also:scale. §g 6o. See also:Professor Fleeming Jenkin's Constant Flow See also:Valve.—In the modules thus far described constant discharge is obtained by varying the See also:area of the orifice through which the water flows. Professor F. Jenkin has contrived a valve in which a constant pressure head is obtained, so that the orifice need not be varied (See also:Roy. See also:Scot. Society of Arts, 1876). Fig. 75 shows a valve of this See also:kind suitable for a 6-in. water See also:main. The water arriving by the main C passes through an See also:equilibrium valve D into the chamber A, and thence through a sluice 0, which can be set for any required area of opening, into the discharging main B. The object of the arrangement is to secure a constant difference of pressure between the See also:chambers A and B, so that a constant discharge flows through the stop valve O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a See also:diaphragm, separating A from a chamber B2 connected by a See also:pipe BI with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and See also:close the equilibrium valve, and conversely a decrease of the difference of pressure will cause the descent of the plunger and open the equilibrium valve wider. Thus a constant difference of pressure is obtained in the chambers A and B. Let w be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square See also:foot, w the See also:weight of the plunger and valve. Then if at any moment pw exceeds w the plunger will rise, and if it is less than w the plunger will descend. Apart from See also:friction, and assuming the valve D to be strictly an equilibrium valve, since w and w are constant, p must be constant also, and equal to w/w. By making w small and w large, the difference of pressure required to ensure the working of the apparatus may be made very small. Valves working with a difference of pressure of s in. of water have been constructed. VI. STEADY FLOW OF COMPRESSIBLE FLUIDS. § 61. See also:External See also:Work during the Expansion of See also:Air.—If air expands without doing any external work, its temperature remains constant. This result was first experimentally demonstrated by J. P. See also:Joule. It leads to the conclusion that, however air changes its See also:state, the See also:internal work done is proportional to the See also:change of temperature. When, in expanding, air does work against an external resistance, either See also:heat must be supplied or the temperature falls. To See also:fix the conditions, suppose 1 lb of air See also:con- fined behind a See also:piston of I sq. ft. area (fig. 76). Let the initial pressure be PI and the volume of the air vi, and suppose this to expand to the pressure p2 and volume v2. If p and v are the corresponding pressure and volume at any intermediate point in the expansion, the work done on the piston during the expansion from v to v+dv is pdv, and the whole work during the expansion from vi to v2, represented by the area See also:abcd, is (v2 pdv. J vi Amongst possible cases two may be selected. See also:Case I.—So much heat is supplied to the air during expansion that the temperature remains constant. Hyperbolic expansion. Then pv = pivi. Work done during expansion per See also:pound of air f vlpdv spiv f vidv/v = plvl See also:log, v2/vi =See also:pith loge pi/p2. (1) Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written (PI/GI) loge GI/G2. (Ia) Then the expansion See also:curve ab is a See also:common See also:hyperbola. Case 2.—No heat is supplied to the air during expansion. Then the air loses an amount of heat equivalent to the external work done and the temperature falls. Adiabatic expansion. In this case it can be shown that Pm' =pivi''+ where -y is the ratio of the specific heats of air at constant pressure and volume. Its value for air is 1.408, and for dry See also:steam 1.135. Work done during expansion per pound of air. = f vipdv = pivi f vidv/vY =—{piETY/(Y—1)}{I/v2Y-I—I/viY—i} = { piviY/ ('y — Olt I /viY-i — I /7121-i } ={pivi/(7—I)}{I—(vi/v2)y-i}. (2) The value of pivi for any given temperature can be found from the data already given. As before, substituting the weights Gi, G2 per cubic foot for the volumes per pound, we get for the work of expansion (pi/GI){ 1/(7 -1)} {I—(G2/G1)7'1, (2a) =pivi{I/('y—1)} [I —(p2/pi)tY-i>/Y}• (2b) § 62. Modification of the Theorem of See also:Bernoulli for the Case of a Compressible Fluid.—In the application of the principle of work to a filament of compressible fluid, the internal work done by the expansion of the fluid, or absorbed in its See also:compression, must be taken into See also:account. Suppose, as before, that AB (fig. 77) comes to A'B' in a See also:short time t. Let pi, wi, vi, GI be the pressure, sectional area of stream, velocity and weight of a cubic foot at A, and p2, See also:w2, v2, G2 the same quantities at B. Then, from the steadiness of motion, the weight of fluid passing A in any given time must be equal to the weight passing B : Giwivit s G2w2v2t. Let zi, z2 be the heights of the sections A and B above any given datum. Then the work of gravity on the See also:mass AB in t seconds is Giwivit (z1 —Z2) = W (zl — z2) t, where W is the weight of See also:gas passing A or B per second. As in the case of an incompressible fluid, the work of the pressures on the ends of the mass AB is pwivit — p2w2v21, = (pi/GI —p2/G2) Wt. The work done by expansion of Wt lb of fluid between A and B is Wt f pdv. The change of kinetic See also:energy as before is (W/2g) (v22—vi2)t. Hence, equating work to change of kinetic energy, W(zi—z2)t+ (pi/Gi—p2/G2)Wt+Wt. vlpdv= (W//2g) (v22—vi2)t; z1+pt/G1 +v12/2g = z2 +p2/G2 +v22/2g —J ~Lpdv. Now the work of expansion per pound of fluid has already given. If the temperature is constant, we get (eq. Ia, § 61) zl+pl/G1+v12/2g=z2+p2/G2 +v22I2g—(pi/G1) loge (Gi/G2)• But at constant temperature pi/G1=p2/G2; :. zl +v12/2g=z2 +v22/2g—(p1/GI) loge (p1/p2), or, neglecting the difference of level, (v22 —v12)/2g = (pi/GI) loge (pi/p2) • (2a) Similarly, if the expansion is adiabatic (eq. 2a, § 61), zi+p1/Gi +vi2/2g = z2+p2/G2 +v22/2g — (p1/Gi) { I /(y - i) } 11 — (p2/pl)iY-06} ; (3) or neglecting the difference of level (v22—v12)/2g=(pi/GI)[I+II(7 —I){—(p2/p1)PY-1)/s}1—p2/G2• (3a) It will be seen hereafter that there is a limit in the ratio pi/p2 beyond which these expressions cease to be true. § 63. Discharge of Air from an Orifice.—The See also:form of the See also:equation of work for a steady stream of compressible fluid is zi +Pi/Gi +vi2/2g = z2 +p2/G2 +v22/2g — (pl/GI) { I / (I, — 1) } A A' B B' (I) been (2) the expansion being adiabatic, because in the flow of the streams of air through an orifice no sensible amount of heat can be communicated from outside. Suppose the air flows from a See also:vessel, where the pressure is pi and the velocity sensibly zero, through an orifice, into a space where the pressure is p2. Let v2 be the velocity of the See also:jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also p2. As air is See also:light, the work of gravity will be small compared with that of the pressures and expansion, so that ziz2 may be neglected. Putting these values in the equation above- p1/Gi =p2/G2+v22/2g–(pi/G1){1/(T–i)} {I—(p2/Pi)(Y–I)/y; v22/2g=p1/Gi–p2/G2+ (p1/G1){I/(7–1)} {i –(p2/pi)(Y–1)/Y} _ (pi/Gi){T/(T -1) —(p2Jpi)Y–1/Y/(T — I)} —p2/G2• Pi/ G17 = p2/G27 .. p2/G2 = (p1/Gi) (p2/pi)(v–1)/Y v22/2g=(p1/Gi){7/(y — I)} {I—(p2/Pi)(1'—1)/Y}; (I) or v22/2g={y/(y- 1) } ((pi/G1) — (p2/G2)); an equation commonly ascribed to L. J. Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by A. J. C. See also:Barre de See also:Saint Venant and L. Wantzel. It has already (§ 9, eq. 4a) been seen that Pi/Gi = (po/Go) (r1/To) where for air po=2116.8, Go=•o8o75 and ro=492'6. v22/2g={ poriy/Goro(1, — 1) } 11 — (P2/P1)(7 — 1)/Y}; (2) or, inserting numerical values, v22/2g = 183.6ri{ 1 — (p2/pi) °'29} ; (2a) which gives the velocity of discharge v2 in terms of the pressure and See also:absolute temperature, pi, ri, in the vessel from which the air flows, and the pressure p2 in the vessel into which it flows. Proceeding now as for liquids, and putting w for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per second at the pressure P2 and temperature See also:r2 is Q2 =cwv2 =cui J [(2gypi/(y– i)G1) (I – (p2/pi)(7–1)/7)] =108.7cw 'I [ri{ I — (p2/p1)°''}] • (3) If the volume discharged is measured at the pressure pi and absolute temperature r1 in the vessel from which the air flows, let Qi be that volume; then p1Qi7 =p2Q2Y; Qi = (p2/pi)1/7Q2 ; Qi=cw 1/ [{2gyp1/(y-I)Gi} 1(p2/pi)2/Y–(p2/pi)(Y+ Let (p2/pi)2/7 —(p2/pi)(7—1)iY=(p2/Pi)1'41—(P2/Pi)1'2=>'; then Qi 1o8•7cw V/ (ri>p). —I) Gil (4) The weight of air at pressure pi and temperature r1 is G1=pi/53.2r1 1b per cubic foot. Hence the weight of air discharged is W = G2Qi =cw d [2gtrp1GV/1/(y— I)] =2'043cwpi J OP/Ti). (5) Weisbach found the following values of the coefficient of discharge c : Conoidal mouthpieces of the form of thel contracted vein with effective pressures c= of •23 to 1.1 See also:atmosphere . . . 0.97 to 0.99 Circular See also:sharp-edged orifices 0.563 „ 0.788 Short cylindrical mouthpieces . . o•81 „ 0.84 The same rounded at the inner end 0.92 „ 0'93 Conical converging mouthpieces . . . 0.90 „ 0.99 § 64. Limit to the Application of the above Formulae.—In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure pi to the pressure P2, while passing from the vessel to the See also:section of the jet considered in estimating the area w. Hence p2 is strictly the pressure in the jet at the See also:plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of See also:simple orifices. Till recently it was tacitly assumed that this pressure p2 was identical with the See also:general pressure external to the orifice. R. D. See also:Napier first discovered that, when the ratio p2/pi exceeded a value which does not greatly differ from o.5, this was no longer true. In that case the expansion of the fluid down to the external pressure is not completed at the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area c.o., of the contracted section of the jet, and may be greater than the area w of the orifice. Napier made experiments with steam which showed that, so See also:long as p2/pi>o•5, the formulae above were trustworthy, when p2 was taken to be the general external pressure, but that, if p2/pi <o•5, then the pressure at the contracted section was See also:independent of the external pressure and equal to 0'5p1. Hence in such cases the constant value 0.5 should be substituted in the formulae for the ratio of the internal and external pressures p2/pi. It is easily deduced from Weisbach's theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio P2/Pi={2/(T+I)}Y 1/Y =0.527 for air =0.58 for dry steam. For a further decrease of external pressure the discharge diminishes, —a result no doubt improbable. The new view of Weisbach's See also:formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish. A. F. Fliegner showed (Civilingenieur xx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the See also:law of flow, as Napier's See also:hypothesis implies, but the curve of flow bends so sharply that Napier's See also:rule may be taken to be a See also:good approximation to the true law. The limiting value of the ratio p2/pi, for which Weisbach's formula, as originally understood, ceases to apply, is for air 0'5767; and this is the number to be substituted for p2/p1 in the formulae when p2/p1 falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuner's See also:paper (Civilingenieur, 1871), and Fliegner's papers (ibid., 1877, 1878). § 65. When a stream of fluid flows over a solid See also:surface, or conversely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the viscosity of the fluid, but generally the See also:laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the solid surface, or by abrupt changes of velocity distributed throughout the fluid; and the energy expended in producing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced. The laws of fluid friction are generally stated thus: 1. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple See also:direct experiment. C. H. See also:Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the See also:rate of decrease of the oscillations was observed. The See also:chief See also:proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures. 2. The frictional resistance of large surfaces is proportional to the area of the surface. 3. At See also:low velocities of not more than i in. per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it flows. At velocities of a ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity. In many See also:treatises on See also:hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much See also:influence on the friction. In Coulomb's experiments a See also:metal surface covered with See also:tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when See also:sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and See also:wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that See also:differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in See also:engineering practice. H. P. G. See also:Darcy's experiments, for instance, showed that in old and incrusted water mains the resistance was twice or some-times thrice as See also:great as in new and clean mains. § 66. See also:Ordinary Expressions for Fluid Friction at Velocities not Extremely Small.—Let f be the frictional resistance estimated in pounds per square foot of surface at a velocity of i ft. per second; w the area of the surface in square feet; and v its velocity in feet per second relatively to the water in which it is immersed. Then, in accordance with the laws stated above, the See also:total resistance of the surface is R = fwv2 (I) where f is a quantity approximately constant for any given surface. If = 2gf /G, R = Gcev2/2g, (2) where E is, like f, nearly constant for a given surface, and is termed the coefficient of friction. The following are See also:average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water. But Coefficient Frictional of Friction, Resistance in _ lb per sq. ft. f New well-painted iron plate . .00489 •00473 Painted and planed See also:plank (Beaufoy) •00350 .00339 Surface of iron See also:ships (See also:Rankine) .00362 •00351 Varnished surface (See also:Froude) . . •00258 .00250 See also:Fine sand surface „ •00418 •00405 Coarser sand surface „ . .00503 •oo488 The distance through which the frictional resistance is overcome is v ft. per second. The work expended in fluid friction is therefore given by the equation Work expended =fwv3 foot-pounds per second (3). = G(.0v3/2g „ ) The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connexion with the cases to which they are related. Many attempts have been made to See also:express the coefficient of friction in a form applicable to low as well as high velocities. The older See also:hydraulic writers considered the resistance termed fluid friction to be made up of two parts,—a See also:part due directly to the distortion of the mass of water and proportional to the velocity of the water relatively to the solid surface, and another part due to kinetic energy imparted to the water striking the roughnesses of the solid surface and proportional to the square of the velocity. Hence they proposed to take in which expression the second See also:term is of greatest importance at very low velocities, and of comparatively little = _ importance at velocities over about z ft.: = per second. Values of t expressed in this _ and similar forms will be given in con- _ = nexion with pipes and canals. — All these expressions must at See also:present be regarded as merely empirical ex- _ —_-:- - pressions serving See also:practical purposes. The frictional resistance will be seen. to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account. § 67. Coulomb's Experiments.—The first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin See also:brass See also:wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given See also:angle, it oscillates under the See also:action of its inertia and the torsion of the wire. The oscillations diminish gradually in consequence of the work done in overcoming the friction of the disk. The diminution furnishes a means of determining the friction. Fig. 78 shows Coulomb's apparatus. LK supports the wire and disk; ag is the brass wire, the torsion of which causes the See also:oscilla- tions; DS is a graduated L_ K disk serving to measure the angles through which the apparatus oscillates. To this the friction disk is rigidly attached See also:hanging in a vessel of water. The friction disks were from 4.7 to 7.7 in. See also:diameter, and they generally made one oscillation in from 20 to 30 seconds, through angles varying from 36o° to 6°. When the velocity of the circumference of the disk was less than 6 in. per second, the resistance was sensibly proportional to the velocity. Beaufol's Experiments.—Towards the end of the 18th See also:century See also:Colonel See also:Mark Beaufoy (1764—1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical and Hydraulic Experiments, See also:London, 183}). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were See also:drawn through water and Mean Resistance in lb per sq. ft. Varnished surface . 2 ft. long 0.41 5o 0.25 Fine sand surface 2 „ 0.81 5o 0.405 This remarkable result is explained thus by Froude: " The portion of surface that goes first in the See also:line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the direction in which it is itself travelling. Consequently the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of lo ft. per second, the difference of resistance, measured on the difference of area, was 0.339 lb per square foot. Also the resistance varied as the I.949th See also:power of the velocity. § 68. Froude's Experiments.—The most important direct experiments on fluid friction at ordinary velocities are those made by See also: The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment were in. thick, 19 in. deep, and from I to 5o ft. in length, cutwater included. A See also:lead See also:keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, See also:varnish, See also:Hay's See also:composition, tinfoil, &c., so as to try the effect of different degrees of roughness of surface. The results obtained by Froude may be summarized as follows: I. The friction per square foot of surface varies very greatly for different surfaces, being generally greater as the sensible roughness of the surface is greater. Thus, when the surface of the board was covered as mentioned below, the resistance for hoards 5o ft. long, at io ft. per second, was Tinfoil or varnish See also:Calico . Fine sand 0.405 Coarser sand . 0.488 „ , 2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 f t. long, For tinfoil the resistance varied as v2•I6. For other surfaces the resistance varied as v2•65. With boards 5o ft. long, For varnish or tinfoil the resistance varied as v1-83. For sand the resistance varied as 3. The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing, the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more sternward. Thus, 0.25 lb per sq. ft. Length of Surface, or Distance from Cutwater, in feet. 2 ft. 8 ft. 20 It. 5o ft. A B C A B C; A B C A B C Varnish 2.00 •41 •390 I.85 •325 .264 I.85 .278 .240 I.83 .250 •226 See also:Paraffin .. •38 .370 1'94 .314 .260 1.93 .271 .237 .. .. Tinfoil 2.16 .30 .295 1.99 •278 .263 1.90 •262 •244 I.83 .246 .232 Calico I•93 .87 •725 I.92 •626 .504 I.89 •531 '447 1.87 -474 -423 Fine sand 2.00 •81 •690 2.00 .583 .450 2.00 •480 •3.84 2'06 .405 .337 See also:Medium sand 2.00 .90 •730 2.00 •625 .488 2.00 .534 .465 2.00 .488 .456 Coarse sand . 2.00 1•IO •880 2.00 •714 .520 2.00 •588 •490 .. the portion of surface which succeeds the first will be rubbing, not sponding to any speed N. From these the values of f and n can he against stationary water, but against water partially moving in its deduced, f being the friction per square foot at unit velocity. For ova direction, and cannot therefore experience so much resistance comparison with Froude's results it is convenient to calculate the from it." resistance at io ft. per second, which is F=fton. § 69. The following table gives a general statement of Froude's The disks were rotated in chambers 22 in. diameter and 3, 6 and results. In all the experiments in this table, the boards had a fine 12 in. deep. In all cases the friction of the disks increased a little cutwater and a fine stern end or run, so that the resistance was as the chamber was made larger. This is probably due to the stilling entirely due to the surface. The table gives the resistances per I of the eddies against the surface of the chamber and the feeding back square foot in pounds, at the See also:standard speed of 600 feet per See also:minute, of the stilled water to the disk. Hence the friction depends not only and the power of the speed to which the friction is proportional, so on the surface of the disk but to some extent on the surface of the that the resistance at other speeds is easily calculated. chamber in which it rotates. If the surface of the chamber is made rougher by covering with coarse sand there is also an increase of resistance. For the smoother surfaces the friction varied as the 1.85th power of the velocity. For the rougher surfaces the power of the velocity to which the resistance was proportional varied from 1.9 to 2.1. This is in agreement with Froude's results. Experiments with a See also:bright brass disk showed that the friction decreased with increase of temperature. The diminution between 41° and 130° F. amounted to 18%. In the general equation M =cN0 for any given disk, ct =0.1328(1 -0.00211) , where et is the value of c for a bright brass disk 0.85 ft. in diameter at a temperature t° F. Columns A give the power of the speed to which the resistance is The disks used were either polished or made rougher by varnish approximately proportional. or by varnish and sand. The following table gives a comparison of Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table. Columns C give the resistance in pounds of a square foot of surface f at the distance sternward from the cutwater stated in the heading. Although these experiments do not directly See also:deal with surfaces of greater length than 50 ft., they indicate what would be the resistances of longer surfaces. For at 50 ft. the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at 10 ft. per second diminishes from 0.41 to 0.32 lb per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250 lb per square foot for an increase from 20 ft. to 5o ft. If the decrease of friction sternwards is due to the See also:generation of a current accompanying the moving plane, there is not at first sight any See also:reason why the decrease should not be greater than that shown by the experiments. The current accompanying the board might be assumed to gain in volume and velocity sternwards, till the velocity was nearly the same as that of the moving plane and the friction per square foot nearly zero. That this does not happen appears to be due to the mixing up of the current with the still water surrounding it. of d Bright brass . m202 to 0.229 Varnish . Additional information and CommentsThere are no comments yet for this article.
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