(1)Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. 1a, § 61)Z1+ P1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) logε(G1/G2).But at constant temperature p1/G1= p2/G2;∴ z1+ v12/2g = z2+ v22/2g − (p1/G1) logε(p1/p2),(2)or, neglecting the difference of level,(v22− v12) / 2g = (p1/G1) logε(p1/p2).(2a)Similarly, if the expansion is adiabatic (eq. 2a, § 61),z1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) {1/(γ − 1) } {1 − (p2/p1)(γ−1)/γ};(3)or, neglecting the difference of level,(v22− v12)/2g = (p1/G1) [1 + 1/(γ − 1) {1 − (p2/p1)(γ−1)/γ)} ] − p2/G2.(3a)It will be seen hereafter that there is a limit in the ratio p1/p2beyond which these expressions cease to be true.§ 63.Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid isz1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1(γ−1)/γ},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 vessel, where the pressure is p1and the velocity sensibly zero, through an orifice, into a space where the pressure is p2. Let v2be the velocity of the 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 light, the work of gravity will be small compared with that of the pressures and expansion, so that z1z2may be neglected. Putting these values in the equation above—p1/G1= p2/G2+ v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ;v22/2g = p1/G1− p2/G2+ (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ}= (p1/G1) {γ/(γ − 1) − (p2/p1)γ−1 /γ/ (γ − 1)} − p2/G2.Butp1/G1γ= p2/G2γ∴ p2/G2= (p1/G1) (p2/p1)(γ−1)/γv22/2g = (p1/G1) {γ/(γ − 1)} {1 − (p2/p1)(γ−1)/γ};(1)orv22/2g = {γ/(γ − 1)} {(p1/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. Barre de Saint Venant and L. Wantzel.It has already (§ 9, eq. 4a) been seen thatp1/G1= (p0/G0) (τ1/τ0)where for air p0= 2116.8, G0= .08075 and τ0= 492.6.v22/2g = {p0τ1γ / G0τ0(γ − 1)} {1 − (p2/p1)(γ−1)/γ};(2)or, inserting numerical values,v22/2g = 183.6τ1{1 − (p2/p1)0.29};(2a)which gives the velocity of discharge v2in terms of the pressure and absolute temperature, p1, τ1, in the vessel from which the air flows, and the pressure p2in the vessel into which it flows.Proceeding now as for liquids, and putting ω for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per second at the pressure p2and temperature τ2isQ2= cωv2= cω √ [(2gγp1/ (γ − 1) G1) (1 − (p2/p1)(γ−1)/γ)]= 108.7cω √ [τ1{1 − (p2/p1)0.29}].(3)If the volume discharged is measured at the pressure p1and absolute temperature τ1in the vessel from which the air flows, let Q1be that volume; thenp1Q1γ= p2Q2γ;Q1= (p2/p1)1/γQ2;Q1= cω √ [ {2gγp1/ (γ − 1) G1} {(p2/p1)2/γ− (p2/p1)(γ+1)/γ}].Let(p2/p1)2/γ− (p2/p1)(γ−1)/γ= (p2/p1)1.41− (p2/p1)1.7= ψ; thenQ1= cω √ [2gγp1ψ / (γ − 1) G1]= 108.7cω √ (τ1ψ).(4)The weight of air at pressure p1and temperature τ1isG1= p1/53.2τ1℔ per cubic foot.Hence the weight of air discharged isW = G1Q1= cω √ [2gγp1G1ψ / (γ − 1)]= 2.043cωp1√ (ψ/τ1).(5)Weisbach found the following values of the coefficient of discharge c:—Conoidal mouthpieces of the form of thecontracted vein with effective pressuresc =of .23 to 1.1 atmosphere0.97to0.99Circular sharp-edged orifices0.563”0.788Short cylindrical mouthpieces0.81”0.84The same rounded at the inner end0.92”0.93Conical converging mouthpieces0.90”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 p1to the pressure p2, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p2is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p2was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p2/p1exceeded a value which does not greatly differ from 0.5, 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 ccω of the contracted section of the jet, and may be greater than the area ω of the orifice. Napier made experiments with steam which showed that, so long as p2/p1> 0.5, the formulae above were trustworthy, when p2was taken to be the general external pressure, but that, if p2/p1< 0.5, then the pressure at the contracted section was 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/p1.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 ratiop2/p1= {2/(γ + 1)}γ−1/γ= 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 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 (Civilingenieurxx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napier’s hypothesis implies, but the curve of flow bends so sharply that Napier’s rule may be taken to be a good approximation to the true law. The limiting value of the ratio p2/p1, 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/p1in the formulae when p2/p1falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuner’s paper (Civilingenieur, 1871), and Fliegner’s papers (ibid., 1877, 1878).VII. FRICTION OF LIQUIDS.§ 65. When a stream of fluid flows over a solid 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 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 direct experiment. C. H. 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 rate of decrease of the oscillations was observed. The chief 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 low velocities of not more than 1 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 of1⁄2ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity.In many treatises on 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 influence on the friction. In Coulomb’s experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when 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 wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. H. P. G. Darcy’s experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains.§ 66.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 1 ft. per second; ω 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 total resistance of the surface isR = fωv2(1)where f is a quantity approximately constant for any given surface. Ifξ = 2gf/G,R = ξGωv2/2g,(2)where ξ is, like f, nearly constant for a given surface, and is termed the coefficient of friction.The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water.Coefficientof Friction,ξFrictionalResistance in℔ per sq. ft.fNew well-painted iron plate.00489.00473Painted and planed plank (Beaufoy).00350.00339Surface of iron ships (Rankine).00362.00351Varnished surface (Froude).00258.00250Fine sand surface (Froude).00418.00405Coarser sand surface (Froude).00503.00488The 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 = fωv3foot-pounds per second= ξGωv3/2g ” ”(3).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 express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid friction to be made up of two parts,—a 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 takeξ = α + β/vin which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about1⁄2ft. per second. Values of ξ expressed in this and similar forms will be given in connexion with pipes and canals.All these expressions must at present be regarded as merely empirical expressions serving 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 brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the 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.Fig. 78 shows Coulomb’s apparatus. LK supports the wire and disk: ag is the brass wire, the torsion of which causes the oscillations; DS is a graduated disk serving to measure the angles through which the apparatus oscillates. To this the friction disk is rigidly attached hanging in a vessel of water. The friction disks were from 4.7 to 7.7 in. diameter, and they generally made one oscillation in from 20 to 30 seconds, through angles varying from 360° 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.Beaufoy’s Experiments.—Towards the end of the 18th century Colonel Mark Beaufoy (1764-1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical and Hydraulic Experiments, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0.339 ℔ per square foot. Also the resistance varied as the 1.949th power of the velocity.§ 68.Froude’s Experiments.—The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810-1879) at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cutwater giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. G is a link by which the resistance of the board is transmitted to a spiral spring H. A bar I rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extension on a greatly increased scale, by a line drawn on the paper cylinder N. This cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage wheels. A second pen O, receiving jerks at every second and a quarter from a clock P, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment were3⁄16in. thick, 19 in. deep, and from 1 to 50 ft. in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hay’s 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:—Fig. 79.1. 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 boards 50 ft. long, at 10 ft. per second, was—Tinfoil or varnish0.25℔ persq. ft.Calico0.47””Fine sand0.405””Coarser sand0.488””2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long,For tinfoil the resistance varied as v2.16.For other surfaces the resistance varied as v2.00.With boards 50 ft. long,For varnish or tinfoil the resistance varied as v1.83.For sand the resistance varied as v2.00.3. 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,Mean Resistance in℔ per sq. ft.Varnished surface2ft. long0.4150”0.25Fine sand surface2”0.8150”0.405This remarkable result is explained thus by Froude: “The portion of surface that goes first in the 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. Consequentlythe portion of surface which succeeds the first will be rubbing, not against stationary water, but against water partially moving in its own direction, and cannot therefore experience so much resistance from it.”§ 69. The following table gives a general statement of Froude’s results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated.Length of Surface, or Distance from Cutwater, in feet.2 ft.8 ft.20 ft.50 ft.ABCABCABCABCVarnish2.00.41.3901.85.325.2641.85.278.2401.83.250.226Paraffin...38.3701.94.314.2601.93.271.237......Tinfoil2.16.30.2951.99.278.2631.90.262.2441.83.246.232Calico1.93.87.7251.92.626.5041.89.531.4471.87.474.423Fine sand2.00.81.6902.00.583.4502.00.480.3842.06.405.337Medium sand2.00.90.7302.00.625.4882.00.534.4652.00.488.456Coarse sand2.001.10.8802.00.714.5202.00.588.490......Columns A give the power of the speed to which the resistance is approximately proportional.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 at the distance sternward from the cutwater stated in the heading.Although these experiments do not directly 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 ℔ per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250 ℔ per square foot for an increase from 20 ft. to 50 ft.If the decrease of friction sternwards is due to the generation of a current accompanying the moving plane, there is not at first sight any 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. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surrounding space, and the current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the current accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also.§ 70.Friction of Rotating Disks.—A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. Experiments carried out by Professor W. C. Unwin (Proc. Inst. Civ. Eng.lxxx.) are useful both as illustrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal pumps. Disks of 10, 15 and 20 in. diameter fixed on a vertical shaft were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three fine wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed.If ω is an element of area on the disk moving with the velocity v, the friction on this element is fωvn, where f and n are constant for any given kind of surface. Let α be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between r and r + dr. Its area is 2πrdr, its velocity αr and the friction of this ring is f2πrdrαnrn. The moment of the friction about the axis of rotation is 2παnfrn+2dr, and the total moment of friction for the two sides of the disk isM = 4παnf∫R0rn+2dr = {4παn/(n + 3) } fRn+3.If N is the number of revolutions per sec.,M = {2n+2πn+1Nn/(n + 3) } fRn+3,and the work expended in rotating the disk isMα = {2n+3πn+2Nn+1/(n + 3) } fRn+3foot ℔ per sec.The experiments give directly the values of M for the disks corresponding to any speed N. From these the values of f and n can be deduced, f being the friction per square foot at unit velocity. For comparison with Froude’s results it is convenient to calculate the resistance at 10 ft. per second, which is F = f10n.The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 in. deep. In all cases the friction of the disks increased a little as the chamber was made larger. This is probably due to the stilling of the eddies against the surface of the chamber and the feeding back of the stilled water to the disk. Hence the friction depends not only on the surface of the disk but to some extent on the surface of the 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 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 = cNnfor any given disk,ct= 0.1328 (1 − 0.0021t),where ctis the value of c for a bright brass disk 0.85 ft. in diameter at a temperature t° F.The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of the results obtained with the disks and Froude’s results on planks 50 ft. long. The values given are the resistances per square foot at 10 ft. per sec.Froude’s Experiments.Disk Experiments.Tinfoil surface0.232Bright brass0.202 to 0.229Varnish0.226Varnish0.220 to 0.233Fine sand0.337Fine sand0.339Medium sand0.456Very coarse sand0.587 to 0.715VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.§ 71. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.Fig. 80.In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction must be equal. Let fig. 80 represent a very short portion of the pipe, of length dl, between cross sections at z and z + dz ft. above any horizontal datum line xx, the pressures at the cross sections being p and p + dp ℔ per square foot. Further, let Q be the volume of flow or discharge of the pipe per second, Ω the area of a normal cross section, and χ the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ ℔, and fall through a height −dz ft. The work done by gravity is then−GQ dz;a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is p − (p + dp) = −dp ℔ per square foot of the cross section. The work of this pressure on the volume Q is−Q dp.The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is χdl.The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3)−ζGχ dl v3/2g,or, since Q = Ωv,−ζG (χ/Ω) Q (v2/2g) dl;the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform,—−GQ dz − Q dp − ζG (χ/Ω) Q (v2/2g) dl = 0.Dividing by GQ,dz + dp/G + ζ (χ/Ω) (v2/2g) dl = 0.Integrating,z + p/G + ζ (χ/Ω) (v2/2g) l = constant.(1)§ 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Thenz1+ p1/G + ζ (χ/Ω) (v2/2g) l1= z2+ p2/G + ζ (χ/Ω) (v2/2g) l2;or, if l2− l1= L, rearranging the terms,ζv2/2g = (1/L) {(z1+ p1/G) − (z2+ p2/G)} Ω/χ.(2)Fig. 81.Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights p1/G and p2/G due to the pressures p1and p2at A and B. Hence (z1+ p1/G) − (z2+ p2/G) is the quantity represented in the figure by DE, the fall of level of the pressure columns, orvirtual fallof the pipe. If there were no friction in the pipe, then by Bernoulli’s equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity DE/AB = h/L is termed the virtual slope of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i.The quantity Ω/χ which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.Introducing these values,ζv2/2g = mh/L = mi.
(1)
Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. 1a, § 61)
Z1+ P1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) logε(G1/G2).
But at constant temperature p1/G1= p2/G2;
∴ z1+ v12/2g = z2+ v22/2g − (p1/G1) logε(p1/p2),
(2)
or, neglecting the difference of level,
(v22− v12) / 2g = (p1/G1) logε(p1/p2).
(2a)
Similarly, if the expansion is adiabatic (eq. 2a, § 61),
z1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) {1/(γ − 1) } {1 − (p2/p1)(γ−1)/γ};
(3)
or, neglecting the difference of level,
(v22− v12)/2g = (p1/G1) [1 + 1/(γ − 1) {1 − (p2/p1)(γ−1)/γ)} ] − p2/G2.
(3a)
It will be seen hereafter that there is a limit in the ratio p1/p2beyond which these expressions cease to be true.
§ 63.Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid is
z1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1(γ−1)/γ},
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 vessel, where the pressure is p1and the velocity sensibly zero, through an orifice, into a space where the pressure is p2. Let v2be the velocity of the 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 light, the work of gravity will be small compared with that of the pressures and expansion, so that z1z2may be neglected. Putting these values in the equation above—
p1/G1= p2/G2+ v22/2g − (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ;
v22/2g = p1/G1− p2/G2+ (p1/G1) {1/(γ − 1)} {1 − (p2/p1)(γ−1)/γ}
= (p1/G1) {γ/(γ − 1) − (p2/p1)γ−1 /γ/ (γ − 1)} − p2/G2.
But
p1/G1γ= p2/G2γ∴ p2/G2= (p1/G1) (p2/p1)(γ−1)/γ
v22/2g = (p1/G1) {γ/(γ − 1)} {1 − (p2/p1)(γ−1)/γ};
(1)
or
v22/2g = {γ/(γ − 1)} {(p1/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. Barre de Saint Venant and L. Wantzel.
It has already (§ 9, eq. 4a) been seen that
p1/G1= (p0/G0) (τ1/τ0)
where for air p0= 2116.8, G0= .08075 and τ0= 492.6.
v22/2g = {p0τ1γ / G0τ0(γ − 1)} {1 − (p2/p1)(γ−1)/γ};
(2)
or, inserting numerical values,
v22/2g = 183.6τ1{1 − (p2/p1)0.29};
(2a)
which gives the velocity of discharge v2in terms of the pressure and absolute temperature, p1, τ1, in the vessel from which the air flows, and the pressure p2in the vessel into which it flows.
Proceeding now as for liquids, and putting ω for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per second at the pressure p2and temperature τ2is
Q2= cωv2= cω √ [(2gγp1/ (γ − 1) G1) (1 − (p2/p1)(γ−1)/γ)]
= 108.7cω √ [τ1{1 − (p2/p1)0.29}].
(3)
If the volume discharged is measured at the pressure p1and absolute temperature τ1in the vessel from which the air flows, let Q1be that volume; then
p1Q1γ= p2Q2γ;
Q1= (p2/p1)1/γQ2;
Q1= cω √ [ {2gγp1/ (γ − 1) G1} {(p2/p1)2/γ− (p2/p1)(γ+1)/γ}].
Let
(p2/p1)2/γ− (p2/p1)(γ−1)/γ= (p2/p1)1.41− (p2/p1)1.7= ψ; then
Q1= cω √ [2gγp1ψ / (γ − 1) G1]
= 108.7cω √ (τ1ψ).
(4)
The weight of air at pressure p1and temperature τ1is
G1= p1/53.2τ1℔ per cubic foot.
Hence the weight of air discharged is
W = G1Q1= cω √ [2gγp1G1ψ / (γ − 1)]
= 2.043cωp1√ (ψ/τ1).
(5)
Weisbach found the following values of the coefficient of discharge c:—
§ 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 p1to the pressure p2, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p2is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p2was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p2/p1exceeded a value which does not greatly differ from 0.5, 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 ccω of the contracted section of the jet, and may be greater than the area ω of the orifice. Napier made experiments with steam which showed that, so long as p2/p1> 0.5, the formulae above were trustworthy, when p2was taken to be the general external pressure, but that, if p2/p1< 0.5, then the pressure at the contracted section was 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/p1.
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/p1= {2/(γ + 1)}γ−1/γ= 0.527 for air= 0.58 for dry steam.
p2/p1= {2/(γ + 1)}γ−1/γ
= 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 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 (Civilingenieurxx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napier’s hypothesis implies, but the curve of flow bends so sharply that Napier’s rule may be taken to be a good approximation to the true law. The limiting value of the ratio p2/p1, 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/p1in the formulae when p2/p1falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuner’s paper (Civilingenieur, 1871), and Fliegner’s papers (ibid., 1877, 1878).
VII. FRICTION OF LIQUIDS.
§ 65. When a stream of fluid flows over a solid 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 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 direct experiment. C. H. 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 rate of decrease of the oscillations was observed. The chief 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 low velocities of not more than 1 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 of1⁄2ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity.
In many treatises on 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 influence on the friction. In Coulomb’s experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when 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 wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. H. P. G. Darcy’s experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains.
§ 66.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 1 ft. per second; ω 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 total resistance of the surface is
R = fωv2
(1)
where f is a quantity approximately constant for any given surface. If
ξ = 2gf/G,
R = ξGωv2/2g,
(2)
where ξ is, like f, nearly constant for a given surface, and is termed the coefficient of friction.
The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water.
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 = fωv3foot-pounds per second= ξGωv3/2g ” ”
(3).
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 express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid friction to be made up of two parts,—a 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
ξ = α + β/v
in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about1⁄2ft. per second. Values of ξ expressed in this and similar forms will be given in connexion with pipes and canals.
All these expressions must at present be regarded as merely empirical expressions serving 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 brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the 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 oscillations; DS is a graduated disk serving to measure the angles through which the apparatus oscillates. To this the friction disk is rigidly attached hanging in a vessel of water. The friction disks were from 4.7 to 7.7 in. diameter, and they generally made one oscillation in from 20 to 30 seconds, through angles varying from 360° 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.
Beaufoy’s Experiments.—Towards the end of the 18th century Colonel Mark Beaufoy (1764-1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical and Hydraulic Experiments, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0.339 ℔ per square foot. Also the resistance varied as the 1.949th power of the velocity.
§ 68.Froude’s Experiments.—The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810-1879) at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cutwater giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. G is a link by which the resistance of the board is transmitted to a spiral spring H. A bar I rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extension on a greatly increased scale, by a line drawn on the paper cylinder N. This cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage wheels. A second pen O, receiving jerks at every second and a quarter from a clock P, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment were3⁄16in. thick, 19 in. deep, and from 1 to 50 ft. in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hay’s 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:—
1. 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 boards 50 ft. long, at 10 ft. per second, was—
2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long,
For tinfoil the resistance varied as v2.16.For other surfaces the resistance varied as v2.00.
For tinfoil the resistance varied as v2.16.
For other surfaces the resistance varied as v2.00.
With boards 50 ft. long,
For varnish or tinfoil the resistance varied as v1.83.For sand the resistance varied as v2.00.
For varnish or tinfoil the resistance varied as v1.83.
For sand the resistance varied as v2.00.
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,
This remarkable result is explained thus by Froude: “The portion of surface that goes first in the 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. Consequentlythe portion of surface which succeeds the first will be rubbing, not against stationary water, but against water partially moving in its own direction, and cannot therefore experience so much resistance from it.”
§ 69. The following table gives a general statement of Froude’s results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated.
Columns A give the power of the speed to which the resistance is approximately proportional.
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 at the distance sternward from the cutwater stated in the heading.
Although these experiments do not directly 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 ℔ per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250 ℔ per square foot for an increase from 20 ft. to 50 ft.
If the decrease of friction sternwards is due to the generation of a current accompanying the moving plane, there is not at first sight any 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. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surrounding space, and the current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the current accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also.
§ 70.Friction of Rotating Disks.—A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. Experiments carried out by Professor W. C. Unwin (Proc. Inst. Civ. Eng.lxxx.) are useful both as illustrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal pumps. Disks of 10, 15 and 20 in. diameter fixed on a vertical shaft were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three fine wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed.
If ω is an element of area on the disk moving with the velocity v, the friction on this element is fωvn, where f and n are constant for any given kind of surface. Let α be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between r and r + dr. Its area is 2πrdr, its velocity αr and the friction of this ring is f2πrdrαnrn. The moment of the friction about the axis of rotation is 2παnfrn+2dr, and the total moment of friction for the two sides of the disk is
M = 4παnf∫R0rn+2dr = {4παn/(n + 3) } fRn+3.
If N is the number of revolutions per sec.,
M = {2n+2πn+1Nn/(n + 3) } fRn+3,
and the work expended in rotating the disk is
Mα = {2n+3πn+2Nn+1/(n + 3) } fRn+3foot ℔ per sec.
The experiments give directly the values of M for the disks corresponding to any speed N. From these the values of f and n can be deduced, f being the friction per square foot at unit velocity. For comparison with Froude’s results it is convenient to calculate the resistance at 10 ft. per second, which is F = f10n.
The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 in. deep. In all cases the friction of the disks increased a little as the chamber was made larger. This is probably due to the stilling of the eddies against the surface of the chamber and the feeding back of the stilled water to the disk. Hence the friction depends not only on the surface of the disk but to some extent on the surface of the 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 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 = cNnfor any given disk,
ct= 0.1328 (1 − 0.0021t),
where ctis the value of c for a bright brass disk 0.85 ft. in diameter at a temperature t° F.
The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of the results obtained with the disks and Froude’s results on planks 50 ft. long. The values given are the resistances per square foot at 10 ft. per sec.
VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.
§ 71. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.
In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction must be equal. Let fig. 80 represent a very short portion of the pipe, of length dl, between cross sections at z and z + dz ft. above any horizontal datum line xx, the pressures at the cross sections being p and p + dp ℔ per square foot. Further, let Q be the volume of flow or discharge of the pipe per second, Ω the area of a normal cross section, and χ the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ ℔, and fall through a height −dz ft. The work done by gravity is then
−GQ dz;
a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is p − (p + dp) = −dp ℔ per square foot of the cross section. The work of this pressure on the volume Q is
−Q dp.
The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is χdl.
The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3)
−ζGχ dl v3/2g,
or, since Q = Ωv,
−ζG (χ/Ω) Q (v2/2g) dl;
the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform,—
−GQ dz − Q dp − ζG (χ/Ω) Q (v2/2g) dl = 0.
Dividing by GQ,
dz + dp/G + ζ (χ/Ω) (v2/2g) dl = 0.
Integrating,
z + p/G + ζ (χ/Ω) (v2/2g) l = constant.
(1)
§ 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Then
z1+ p1/G + ζ (χ/Ω) (v2/2g) l1= z2+ p2/G + ζ (χ/Ω) (v2/2g) l2;
or, if l2− l1= L, rearranging the terms,
ζv2/2g = (1/L) {(z1+ p1/G) − (z2+ p2/G)} Ω/χ.
(2)
Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights p1/G and p2/G due to the pressures p1and p2at A and B. Hence (z1+ p1/G) − (z2+ p2/G) is the quantity represented in the figure by DE, the fall of level of the pressure columns, orvirtual fallof the pipe. If there were no friction in the pipe, then by Bernoulli’s equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity DE/AB = h/L is termed the virtual slope of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i.
The quantity Ω/χ which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.
Introducing these values,
ζv2/2g = mh/L = mi.