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Originally appearing in Volume V14, Page 45 of the 1911 Encyclopedia Britannica.
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DISTRIBUTION OF ENERGY IN INCOMPRESSIBLE FLUIDS. § 29. Application of the Principle of the Conservation of Energy to Cases of Stream Line Motion.--The external and internal work done on a mass is equal to the change of kinetic energy produced. In many hydraulic questions this principle is difficult to apply, be-cause from the complicated nature of the motion produced it is difficult to estimate the total kinetic energy generated, and because in some cases the internal work done in overcoming frictional or viscous resistances cannot be ascertained; but in the case of stream line motion it furnishes a simple and important result known as Bernoulli's theorem. Let AB (fig. 25) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motion, AB is a fixed path in space through which a stream of fluid is constantly flowing. Let 00 be the free surface and XX any horizontal datum line. Let 0 0 A A' -1-- co be the area of a normal cross section, v the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and wi, PI, vi, zi the same quantities at B. Suppose that in a short time t the mass of fluid initially occupying AB comes to A'B'. Then AA', BB' are equal to vt, vit, and the volumes of fluid AA', BB' are the equal inflow and outflow =Qt=wvt=colvit, in the given time. If we suppose the filament AB surrounded by other filaments moving with not very different velocities, the frictional or viscous resistance on its surface will be small enough to be neglected, and if the fluid is incompressible no internal work is done in change of volume. Then the work done by external forces will be equal to the kinetic energy produced in the time considered. The normal pressures on the surface of the mass (excluding the ends A, B) are at each point normal to the direction of motion, and do no work. Hence the only external forces to be reckoned are gravity and the pressures on the ends of the stream. The work of gravity when AB falls to A'B' is the same as that of transferring AA' to BB'; that is, GQt (z—zi). The work of the pressures on the ends, reckoning that- at B negative, because it is opposite to the direction of motion, is (Pw Xvt) — (Piwt Xvit) = Qt(p—pi). The change of kinetic energy in the time t is the difference of the kinetic energy originally possessed by AA' and that finally acquired by BB', for in the intermediate part A'B there is no change of kinetic energy, in consequence of the steadiness of the motion. But the mass of AA' and BB' is GQt/g, and the change of kinetic energy is therefore (GQt/g) (v12/2 —v2(2). Equating this to the work done on the mass AB, GQt(z—zi)+Qt(p—pi) = (GQt/g) (v12/2 —v2/2). Dividing by GQt and rearranging the terms, v2/2g+P/G +z=v12/2g+pl/G+zi; (I) or, as A and B are any two points, v2/2g+p/G+z =constant = H. (2) Now v2/2g is the head due to the velocity v, p/G is the head equivalent to the pressure, and z is the elevation above the datum (see § 16). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament, z is easily seen to be the work in foot-pounds which would be done by i lb of fluid falling to the datum line, and similarly p/G and v2/2g are the quantities of work which would he done by I lb of fluid due to the pressure p and velocity v. The expression on the left of the equation is, therefore, the total energy of the stream at the section considered, per lb of fluid, estimated with reference to the C. '594 'SM > datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total energy per lb of fluid is uniformly distributed along the stream line. If the free surface of the fluid 00 is taken as the datum, and -h, -hi are the depths of A and B measured down from the free surface, the equation takes the form v2/2g+p/G - h =vie/2g+pi/G -hi ; (3) v2/2g -1- p/G - h = constant. (3a) § 30. Second Form of the Theorem of Bernoulli.—Suppose at the two sections A, B (fig. 26) of an elementary stream small vertical pipes are introduced, which may be termed pressure columns projected surface as HI, and the pressures parallel to the axis of the pipe, normal to these projected surfaces, balance each other. Similarly the pressures on BC, CD balance those on GH, EG. In the same way, in any combination of enlargements and contractions, a balance of pressures, due to the flow of liquid parallel to the or generally c 5 ---- FIG. 26. . (§ 8), having their lower ends accurately parallel to the direction of flew. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence b=p/G, and b'=p,/G. Consequently the tops of the pressure columns A' and B' will be at total heights b c = p/G+z and b'+c' =pi/G-l-zi above the datum line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore X~- -- ~- X ` axis of the pipe, will be found, provided the sectional area and direction of the ends are the same. The following experiment is interesting. Two cisterns provided with converging pipes were placed so that the jet from one was ex-I opposite the entrance to the other. The cisterns being filled ya --__= - a 33 ---I--- ~ _ _ H = a _( —PI) /G+(z; and this by equation (I), § 29 is (vie-v2)/2g. That is, the fall of ®® free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line A'B' is sometimes called the line of hydraulic gradient, though this term is also used in cases where friction needs to be taken into account. It is the line the height of which above datum is the sum of the elevation and pressure head at that point, and it falls below a horizontal line A"B" drawn at H ft. above XX by the quantities a=v2/2g and a'=vie/2g, when friction is absent. very nearly to the same level, the jet from the left-hand cistern A § 31. Illustrations of the Theorem of Bernoulli. In a lecture to entered the right-hand cistern B (fig. 31), shooting across the free the mechanical section of the British Association in 1875, W. Froude space between them without any waste, except that due to indirect-gave some experimental illustrations of the principle of Bernoulli. ness of aim and want of exact correspondence in the form of the He remarked that it was a common but erroneous impression that orifices. In the actual experiment there was 18 in. of head in the a fluid exercises in a contracting pipe A (fig. 27) an excess of pressure right and 202 in. of head in the left-hand cistern, so that about against the entire converging surface which it meets, and that, conversely, A as it enters an enlargement B, a relief of pressure is experienced by the - entire diverging surface of the pipe. '' Further it is commonly assumed that when passing through a contraction A y C, there is in the narrow neck an FIG. 30. excess of pressure due to the squeezing together of the liquid at that 22 in. were wasted in friction. It will be seen that in the open space point. These impressions are in no respect correct; the pressure between the orifices there was no pressure, except the atmospheric is smaller as the section of the pipe is smaller and conversely. pressure acting uniformly throughout the system. Fig. 28 shows a pipe so formed that a contraction is followed by § 32. Ventur^ Meter.—An ingenious application of the variation an enlargement, and fig. 29 one in which an enlargement is followed of pressure a-.d velocity In a converging and diverging pipe has been by a contraction. The B vertical pressure columns show the decrease of pressure at the contrac- tion and increase of pressure at the enlarge- ment. The line abc in both figures shows the variation of free surface level, supposing the pipe frictionless. In actual pipes, however, work is expended in friction against the pipe; the total head diminishes in proceeding along the pipe, and the free surface level is a line such as abici, falling below abc. Froude further pointed out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part made by Clemens Herschel in the construction of what he terms a exactly balances the resultant pressure on the diverging part so I Venturi Meter for measuring the flow in water mains. Suppose that, that there is no tendency to move the pipe bodily when water flows as iii fig. 32, a contraction is made in a water main, the change of through it. Thus the conical part AB (fig. 30) presents the same section being gradual to avoid the production of eddies. The ratio p A C of the cross sections at A and B, that is at inlet and throat, is in actual meters 5 to I to 20 to I, and is very carefully determined by the maker of the meter. Then, if v and u are the velocities at A and B, •u=pv. Let pressure pipes be introduced at A, B and C, 1_ Dalbm Lin_ . and let Hi, H, H2 be the pressure heads at those points. Since the velocity at B is greater than at A the pressure will be less. Neglecting friction Hi +v2/2g -- H +u,/2g, H,–H = (u2—v2)/2g = (p2–1)v2/2g. Let h = H1–H be termed the Venturi head, then u = {p2.2gh/(p2–I)}, from which the velocity through the throat and the discharge of the main can be calculated if the areas at A and B are known and h observed. Thus if the diameters at A and B are 4 and I2 in., the areas are 12.57 and 113.1 sq. in., and p=9, u = ' 81/80./ (2gh) =1.007 V (2gh). If the observed Venturi head is 12 ft., u =28 ft. per sec., and the discharge of the main is 28 X12.57 =351 cub. ft. per sec. Hence by a simple observation of pressure difference, the flow in the main .at any moment can be determined. Notice that the pressure height at C will be the same as at A except for a small loss h1 due to friction and eddying between A and B. To get the pressure at the throat very exactly Herschel surrounds it by an annular passage communicating with the throat by several small holes, sometimes formed in vulcanite to prevent corrosion. Though con- structed to prevent eddying as much as possiule there is some eddy loss The main effect of this is to cause a loss of head between A and C which may vary from a fraction of a foot to perhaps 5 ft. at the highest velocities at which a meter can be used. The eddying also affects a little the Venturi head h. Consequently an experi- mental coefficient must be determined for each meter by tank measure- ment. The range of this coefficient is, however, surprisingly small. If to allow for friction, u=ki) (2gh), then Herschel found values of k from 0.97 to 1.0 for throat velocities varying from 8 to 28 ft. per sec. The meter is extremely convenient. At Staines reservoirs there are two meters of this type on mains 94 in. in diameter. Herschel contrived a recording arrangement which records the variation of flow from hour to hour and also the total flow in any given time. In Great Britain .be meter is constructed by G. Kent, who has made improvements Inlet in the recording arrangement. In the Deacon Waste Water Meter (fig. 33) a different principle is used. A disk D, partly counter-balanced by a weight, is suspended in the water flowing through the main in a conical chamber. The unbalanced weight of the disk is supported by the impact of the water. If the discharge of the main increases the disk rises, but as it rises its position in the chamber is such that in consequence of the larger area the velocity is less. It finds, therefore, a new position of equilibrium. A pencil P records on a drum moved by clockwork the position of the disk, and from this the variation of flow is in- ferred. § 33. Pressure, Velocity and Energy in Different Stream Lines.—The equation of Bernoulli gives the variation of pressure and velocityfrom point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear.) Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines p+dp f'' -0'11 O' making an angle ¢ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ = ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of fluid, H =z+p/G -{-v2/2g. Differentiating, we get dH = dz+dp/G+vdv/g, for the increment of energy between Q and P. But dz=PQ cos 4)=ds cos 4); ...dH =dp/G+vdv/g+ds cos 0, (la) where the last term disappears if the motion is in a horizontal plane. Now imagine a small cylinder of section w described round PQ as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is wds and its weight Gwds. Hence, taking the components of the forces parallel to PQ wdp=Gv2wds/gp–Gw cos r¢ds, where p is the radius of curvature of the stream line at Q. Consequently, introducing these values in (I), dH = v2ds/gp +vdv/g = (v/g) (v/p +dv/ds)ds. (2) CURRENTS § 34• Rectilinear Current.—Suppose the motion is in parallel straight stream lines (fig. 35) in a vertical plane. Then p is infinite, and from eq. (2), § 33, dH =vdv/g. Comparing this with (I) we see that dz+dp/G=o; z +p/G = constant ; (3) or the pressure varies hydrostatically as in a fluid at rest. For two stream lines in a horizontal P plane, z is constant, and there-fore p is constant. Radiating Current.—Suppose water flowing radially between horizontal parallel planes, at a distance apart =S. Conceive two cylindrical sections of the current at radii r, and r2, where the velocities are v, and v2, and the pressures pi and p2. Since the flow across each cylindrical section of the current is the same, Q = 2lrr15vi = 27rr257)2 nu' = r2v2 r,/r2 =v2/vr. (4) 1 The following theorem is taken from a paper by J. H. Cotterill, " On the Distribution of Energy in a Mass of Fluid in Steady Motion," Phil. Mag., February 1876. Outlet (I) dz 1 Q T , Fhe velocity would be infinite at radius o, if the current could be conceived to extend to the axis. Now, if the motion is steady, H = p1/G +v1'-/2g = p2/G +v22/2g ; = p2/G -f- r,2vr/r222g ; (p2p,)/G =er'(I–rig/r22)leg; P2/(-; = H–r,2ei2/r222g. Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 36, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xy2=c3. This curve is asymptotic to a horizontal line, H ft. above the line from which the pressures are measured, and to the axis of the current. Free Circular Vortex.—A free circular vortex is a revolving mass )f water, in which the stream lines are concentric circles, and in which 14,14 --- I I 7- - I II i ft 1 IG § 35. Forced Vortex.—If the law of motion in a rotating current is different from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), § 33, considering two circular stream lines of radii r and r+dr (%g. 37), we have p=r, ds = dr. If the angular velocity is a, then v = ar and dv = adr. Hence dH = a2rdr/g + a2rdr/g = 2 a2rdr/g. Comparing this with (I), § 33, and putting dz=o, because the motion is horizontal, dp/G -f - a2rdr/g = 2 a2rdr/g, dp/G = a°rdr/g, p!G =a2r2/2g+constant. (9) Let p,, r,, v1 be the pressure, radius and velocity of one cylindrical section, p2, r2, v2 those of another; then p,/G-a2r12/2g = p2/G–a2r22/2g ; (p2–p1)/G = a2(r22-ri2)/2g = (2'22-v,2)/2g• (Io) That is, the pressure increases from within outwards in a curve (5) (6) the total head for each stream lire is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities propel to their new positions under the action of the existing fluid pressures only. For such a current, the motion being horizontal, we have for all the circular elementary streams I / H = p/G +v2/2g = constant ; dH = dp/G +vdv/g = o. (7) Consider two stream lines at radii r and r+dr (fig. 36). Then in (2), § 33, p=r and ds=dr, v2dr/gr-hvdv/g = o, dv/v =–dr/r, v = I/r, (8) precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case. Free Spiral Vortex.—As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any pro-portions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of tl.e current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal pump or turbine into a chamber, it forms a free vortex of this kind. The water flows spirally outwards, its velocity diminishing and its which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (ag. 37).

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