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Originally appearing in Volume V14, Page 69 of the 1911 Encyclopedia Britannica.
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Q00 0 .984 See also:

loo° 1.26o 110° 1.556 120° 1.861 130° 2.158 140° 2.431 =0.131 -}-1.847 (d/2p) r, (I) (4a) value of d by assuming if the See also:average demand is 25 gallons per See also:head per See also:day, the mails should be calculated for 50 gallons per head per day. § 86. Determination of the Diameters of Different Parts of a See also:Water See also:Main.—When the See also:plan of the arrangement of mains is determined upon, and the See also:supply to each locality and the pressure required is ascertained, it remains to determine the diameters of the pipes. Let fig. 97 show an See also:elevation of a main See also:ABCD..., R being the See also:reservoir from which the supply is derived. Let NN be the datum See also:line of the levelling operations, and Ha, Hb...the heights of the main above the datum line, Hr being the height of the water See also:surface in the (b) A second method is to obtain a rough '=a. This value is d' V (32Q2/gr2i)V a =o.6319 V, (Q2/i)V a. Then a very approximate value of 1• is =a(I+I/I2d'); and a revised value of d, not sensibly differing from the exact value, is d" =V (32Q2; gr2i)V i'' °0.6319 V (Q2/i)V r'. (c) See also:Equation 7 may be put in the See also:form d =V (32aQ2/gr2i) V (1+111 ad). (9) Expanding the See also:term in brackets, _ tE E 11 (I+I/12d) =1+I/6od—I/1800d2... Neglecting the terms after the second, d=J(32a!gr2)V(Q2/i).(I+I/6od) X z =V(32a/gr2)V(Q2/2)+o.o1667;(9a) il'° b and V (32a/gr2) =0.219 for new pipes s =o•252forincrusted pipes. § 85.

Arrangement of Water Mains 4'e for Towns' Supply.—See also:

Town mains are •-.-- --..P-' n-~.Txs-.°-' : r See also:ea- w .:_..~ ~,. usually suppliedby See also:gravitation from FIG. 98. a service reservoir, which in turn is supplied by gravitation from a storage reservoir or by pumping from a See also:lower level. The service reservoir should contain three days' supply or in important cases much more. Its elevation should he such that water is delivered at a pressure of at least about too ft. to the highest parts of the See also:district. The greatest pressure in the mains is usually about 200 ft., the pressure for which See also:ordinary p)aes and fittings are designed. Hence if the district supplied has See also:great See also:variations of level it must be divided into zones of higher and lower pressure. Fig. 96 shows a district of two zones each with its service reservoir and a range of pressure in the lower district from loo to 200 ft. The See also:total supply required is in See also:England about 25 gallons per head per day. But in many towns, and especially in See also:America, the supply is considerably greater, but also in many cases a See also:good See also:deal of the supply is lost by leakage of the mains.

The supply through the See also:

branch mains of a distributing See also:system is calculated from the See also:population supplied. But in determining the capacity of the mains the fluctuation of the demand must be allowed for. It is usual to take the maximum demand at twice the average demand. Hence reservoir from the same datum. Set up next heights AA, BB1,... representing the minimum pressure height necessary for the adequate supply of each locality. Then A1B1C1D1... is a line which should form a lower limit to the line of virtual slope. Then if heights I)0, 1)b, I)c... are taken representing the actual losses of head in each length lo, lb, la... of the main, AoBoCo will be the line of virtual slope, and it will be obvious at what points such as Do and Eo, the pressure is deficient, and a different choice of See also:diameter of main is required. For any point z in the length of the main, we have Pressure height =Hr—Hz—Oa.+b+- • .~s)• Where no other circumstance limits the loss of head to be assigned to a given length of main, a See also:consideration of the safety of the main from fracture by See also:hydraulic See also:shock leads to a See also:limitation of the velocity of flow. Generally the velocity in water mains lies between I i and 42 ft. per second. Occasionally the velocity in pipes reaches so ft. per second, and in hydraulic machinery working under enormous pressures even 20 ft. per second. Usually the velocity diminishes along the main as the See also:discharge diminishes, so as to reduce somewhat the total loss of head which is liable to render the pressure insufficient at the end of the main. J.

T. Fanning gives the following velocities as suitable in pipes for towns' supply: Diameter in inches 4 8 12 18 24 30 36 Velocity in feet per sec. . . 2.5 3'0 3'5 4'5 5'3 6.2 7.0 § 87. Branched See also:

Pipe connecting Reservoirs at Different Levels.—Let A, B, C (fig. 98) be three reservoirs connected by the arrangement of pipes shown,—l1, d1, Qi, v1; 12, d2, Q2, v2; l3, d3, Q3, v3 being the length, diameter, discharge and velocity in the three portions of the main pipe. Suppose the dimensions and positions of the pipes known and the discharges required. If a pressure See also:column is introduced at X, the water will rise to a height XR, measuring the pressure at X, and aR, Rb, See also:Rc will be the lines of virtual slope. If the See also:free surface level at R is above b, the reservoir A supplies B and C, and if R is below b, A and B supply C. Consequently there are three cases: I. R above b; Q1=Q2+Q3. II.

R level with b; Q1=Q3; Q2=o To determine which See also:

case has to be dealt with in the given conditions, suppose the pipe from X to B closed by a sluice. Then there is a See also:simple main, and the height of free surface h' at X can be determined. For this See also:condition ha —h' = I'(v12/2g)(411/d1) = 323"Q'2l1/gr2d15 ; Ii'—ha =I-(v32/2g)(4l3/d3) =32f'Q'2l3/gr2d35; where Q' is the See also:common discharge of the two portions of the pipe. Hence (ha—h')/(h'—ha) =11d35/13d15, from which h' is easily obtained. If then h' is greater than hb, opening the sluice between X and B will allow flow towards B, and the case in See also:hand is case I. If h' is less than h5, opening the sluice will allow flow from B, and the case is case III. If h'=hb, the case is case II., and is already completely solved. 4•---~-_Nigh See also:eve/• See also:Zone 1 i See also:Low Level Zone=•-~_. 1 _ soi zo6 I -`I > orssi Fm. 97. The true value of h must See also:lie between h' and hb. Choose a new value of h, and recalculate QI, Q2, Q3.

Then if Q,>Q2+Q3 in case I., or Q,+Q2>Q3 in case III., the value chosen for h is too small, and a new value must be chosen. If Qi <Qs+Q3 in case I., or Qi+Qi <Q3 in case III., the value of h is too great. Since the limits between which h can vary are in See also:

practical cases not very distant, it is easy to approximate to values sufficiently accurate. § 88. Water See also:hammer.—If in a pipe through which water is flowing a sluice is suddenly closed so as to See also:arrest the forward See also:movement of the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This See also:action is termed water hammer or water See also:ram. The fluctuation of pressure is an oscillating one and gradually See also:dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J. Joukowsky at See also:Moscow in 1898 (See also:Stoss in Wasserleitungen, St See also:Petersburg, 1900), and the results are generally confirmed by experiments made by E. B.

See also:

Weston and R. C. See also:Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in. diameter, from woo to 2500 ft. in length. The sluice closed in 0.03 second, and the fluctuations of pressure were automatically registered. The maximum excess pressure due to water-hammer action was as follows: Pipe 4-in. diameter. Pipe 6-in. diameter. Velocity Excess Pressure. ft. per Velocity Excess Pressure. ft. per sec. lb per sq. in. sec. lb per sq. in. 0.5 -- 31 I 0.6 43 2.9 168 3.0 173 4.1 232 5.6 369 9?- -~ 519 7 5 ¢26 In sonic cases, in fixing the thickness of water mains, too lb per sq. in. excess pressure is allowed to See also:cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable See also:allowance (see also Carpenter, Am.

See also:

Soc. Mech. Eng., 1893). IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES § 89. Flow of See also:Air in See also:Long Pipes.—When air flows through a long pipe, by far the greater See also:part of the See also:work expended is used in over-coining frictional resistances due to the surface of the pipe. The work expended in See also:friction generates See also:heat, which for the most part must be See also:developed in and given back to the air. Some heat may be transmitted through the sides of the pipe to surrounding materials, but ,n experiments hitherto made the amount so conducted away appears to be very small, and if no heat is transmitted the air in the See also:tube must remain sensibly at the same temperature during expansion. In other words, the expansion may be regarded as isothermal expansion, the hear generated by friction exactly neutralizing the cooling due to the work done. Experiments on the pneumatic tubes used for the transmission of messages. by R. S. Gulley and R.

See also:

Sabine (Prot. Inst. Cie. Eng. xliii.), show that the See also:change of temperature of the air flowing along the tube is much less than it would be in adiabatic expansion. § 90. See also:Differential Equation of the Steady See also:Motion of Air Flowing in a Long Pipe of See also:Uniform See also:Section.—When air expands at a See also:constant See also:absolute temperature r, the relation between the pressure p in pounds per square See also:foot and the See also:density or See also:weight per cubic foot G is given by the equationtime dt the See also:mass of air between AoA, comes to A'oA'I so that AoA'o = udt and AiA'i = (u+du)dti. Let tf be the section, and m the hydraulic mean See also:radius of the pipe, and W the weight of air flowing through the pipe per second. From the steadiness of the motion the weight of air between the sections AoA'o, and A,A'I is the same. That is, Wdt = GSludt = GSl (u+du)dt. By See also:analogy with liquids the head lost in friction is, for the length dl (see § 72, eq. 3), I-(u2/2g)(dl/m). Let H=u'°/2g.

Then the head lost is '(H/m)dl; and, since Wdt lb of air flow through the pipe in the See also:

time considered, the work expended in friction is —i'(H/m)Wdl dt. The change of kinetic See also:energy in dt seconds is the difference of the kinetic energy of A,A'o and See also:ALA',, that is, (W/g)dt{ (u+du)'—u2}/2 = (See also:Wig) udu dt =WdHdt. The work of 'expansion when Sludt cub. ft. of air at a pressure p expand to tl(u+du)dt cub. ft. is Slpdudt. But from (3a) u=crW/Slp, and therefore du/dp = —crW /S2p2. And the work done by expansion is— (crW/ p)dp dl. The work done by gravity on the mass between Ao and Ai is zero if the pipe is See also:horizontal, and may in other cases be neglected without great See also:error. The work of the pressures at the sections AoAI is ptludt — (p+dp)See also:f2(u+du) dt = — (pdu+udp)Sldt pure-constant, Pdu+udp=o, and the work of the pressures is zero. Adding together the quantities of work, and equating them to the change of kinetic energy, WdHdt = — (crW /p)dp dt —I'(H/m)W dl dt dH + (cr/p)d p+DH/m)dl =o, dH/H +(cr/Hp)dp+(-dl/m =o But u =crW /Slp, and H = u'/2g = c'r2W 2/2g922p2, :.dH/H • (2gtf'p/crW''~)dpd 1dl/m o. (4a) For tubes of uniform section m is constant; for steady motion kV is constant; and for isothermal expansion r is constant. Integrating, See also:log H +gSf'p2/W2cr+I'l/m = constant; (5) for l=o, let H=Ho, and p=po; and for 1=1, let H =Hi, and p =Pi. log (H,/IIo)-I-(g~2/W2cr) (p,2-po-')+ /m=o, (5a) where po is the greater pressure and pi the less, and the flow is from Ao towards Ai. By replacing W and H, log(po/PI)+(gcr/uo'po2)(pi2—Po'')+I"llm=o.

(6) Hence the initial velocity in the pipe is no—J[{gcr(pot—p,2))/{po2(Il/m+log(po/PI)}]. (7) When 1 is great, log po/p, is comparatively small, and then no=,/ [(germ/i"l){(p 2—p,2)/to2}], (7a) a very simple and easily used expression. For pipes of circular section m=d/4, where d is the diameter: uo-[(gcrd/4Il){(po2—p,2)/po21]; (7b) or approximately uo=(1.1319—0.7264p,/po)vt (nerd/4I"l)• (7c) § 91. Coefficient of Friction for A-ir.—A discussion by See also:

Professor Unwin of the experiments by Culley and Sabine on the See also:rate of transmission of See also:light See also:carriers through pneumatic tubes, in which there is steady flow of air not sensibly affected by any resistances other than surface friction, furnished the value I-=•007. The pipes were See also:lead pipes, slightly moist, 21 in. (0.187 ft.) in diameter, and in lengths of 2000 to nearly 600o ft. In some experiments on the flow of air through See also:cast-See also:iron pipes A. See also:Arson found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting I•=a/v-K8, he obtained the following values Diameter of Pipe a I• for loo ft. in feet. per second. 1.64 •00129 '00483 .00484 1.07 •00972 •00640 •00650 '83 .01525 .00704 00719 338 •03604 •00941 •00977 I •266 ' •03790 '00959 '00997 L_164 04518 OI167 012I2 It is See also:worth while to try if these See also:numbers can be expressed in the form proposed by See also:Darcy for water.

For a velocity of loo ft. per second, and without much error for higher velocities, these numbers agree fairly with the See also:

formula I'=O.005(I+3/I0d), (9) which only differs from Darcy's value for water in that the second term, which is always small except for very small pipes, is larger. p/G=cr, (1) where c=53.15. Taking r=521, corresponding to a temperature of 6o° Fahr., CT =27690 foot-pounds. (2) The equation of continuity, which expresses the condition that in steady motion the same weight of fluid, W, must pass through each See also:cross section of the stream in the unit of time, is CSlu=W=constant, (3) where tl is the section of the pipe and u the velocity of the air. Combining (I) and (3). Slup/W%=cr =constant. (3a) 41'2 Since the work done by gravity on the air during its flow through a pipe due to variations of its level is generally small compared with the work done by changes of pressure, the former may in many cases be neglected. Consider a See also:short length dl of the pipe limited by sections A0, AI at a distance dl (fig. 94). Let p, u be the pressure and velocity at Ao, p+dp and u+du those at A,. Further, suppose that in a very short ;< L- di •tee i Ao Ao Ai But from (3a) (4) (8) Some later experiments on a very large See also:scale, by E. Stockalper at the St Gotthard See also:Tunnel, agree better with the value =0.0028(I+3/See also:tod).

These pipes were probably less rough than Arson's. When the variation of pressure is very small, it is no longer safe to neglect the variation of level of the pipe. For that case we may neglect the work done by expansion, and then zo —z1—po/Go —pl/Gl --(v2/2g) (1/m) = o, (Io) precisely See also:

equivalent to the equation for the flow of water, zo and zt being the elevations of the two ends of the pipe above any datum, po and pi the pressures, Go and GI the densities, and v the mean velocity in the pipe. This equation may be used for the flow of See also:coal See also:gas. § 92. See also:Distribution of Pressure in a Pipe in which Air is Flowing.—From equation (7a) it results that the pressure p, at 1 ft. from that end of the pipe where the pressure is po, is p=poJ(T —'luo2/mgcr); (II) which is of the form p=J (al+b) for any given pipe with given end pressures. The See also:curve of free surface level for the pipe is, therefore, a See also:parabola with horizontal See also:axis. Fig. Too shows calculated curves of pressure for two of Sabine's experiments, in one of which the pressure was greater than atmo- 4 8.0 2113.5 Ft. 922'7 t. 63+o-.s t.. a4S4Ft. spheric pressure, and in the other less than atmospheric pressure.

The observed pressures are given in brackets and the calculated pressures without brackets. The pipe was the pneumatic tube between Fenchurch See also:

Street and the Central Station, 2818 yds. in length. The pressures are given in inches of See also:mercury. Variation of Velocity in the Pipe.—Let po, uo be the pressure and velocity at a given section of the pipe; p, u, the pressure and velocity at any other section. From equation (3a) up= crW/S2 = constant ; so that, for any given uniform pipe, up =11opo, u=uopo/p; (12) which gives the velocity at any section in terms of the pressure, which has already been determined. Fig. See also:lot gives the velocity curves for the two experiments of Culley and Sabine, for which the pressure curves have already been See also:drawn. It will be seen that the velocity increases considerably towards that end of the pipe where the pressure is least. § 93• 'Weight of Air Flowing per Second.—The weight of air discharged per second is (equation 3a) W =Sluopo/cr. From equation (7b), for a pipe of circular section and diameter d, W = 4ir'1 (gd5(po2—pi2) /rlcrl, =•61I-1Id5(p02 See also:pie)/ilr}. (13) Approximately W = (•6916po—•443811) (d5/f'lr)i. (13a) § 94. Application to the Case of Pneumatic Tubes for the Trans-See also:mission of Messages.—In See also:Paris, See also:Berlin, See also:London, and other towns, it has been found cheaper to transmit messages in pneumatic tubesthan to See also:telegraph by See also:electricity.

The tubes are laid underground with easy curves; the messages are made into a See also:

roll and placed in a light See also:felt See also:carrier, the resistance of which in the tubes in Lor_don is only ; oz. A current of air forced into the tube or drawn through it propels the carrier. In most systems the current of air is steady and continuous, and the carriers are introduced or removed without materially altering the flow of air. Time of Transit through the Tube.—Putting t for the time of transit from o to 1, 1 t= odl/u, From (4a) neglecting dH/H, and putting m=d/4, dl = gdS22 pdp/25 \V2cr. u=Wcr/pf2; dl/u(=gdc23p2dp/2i W3c2r2; t =J Plgdirp''dp/20,V3C2r2, = gd[23 (po3_pi3) /61 `V3c2r2• W = pouol/cr ; t = gdcr (po3—p13) /631'0143, = l (po 3—p13) /6 (gcrd) i (p02—p12) i. If r=521°, corresponding to 6o° F.,/ t=•oo1412Onn(p03—p13(po2—p12)1; (15a) which gives the time of transmission in terms ofL the initial and final pressures and the dimensions of the tube. Mean Velocity of Transmission.—The mean velocity is l/t; or, for 1. =521°, { 14meau = o.7o8 d (p02 . p12) i /i-1 (p03 — P13)1. The following table gives some results: Ab^ures Mean Velocities for Tubes of a. Pressures in lb per sq. in. length in feet. PO Pr 1000 2000 3000 4000 5000 Vacuum 15 5 99'4 70'3 57'4 49'7 44'5 Working((.

15 10 67.2 47'5 38.8 34'4 30'1 Pressure 1 ' . 2O 15 57.2 40.5 33.0 28.6 25.6 Working 2S 15 74'6 52'7 43 I 37 3 33'3 30 15 84.7 6o•o 49.0 42.4 37.9 Limiting Velocity in the Pipe when the Pressure at one End is diminished indefinitely.—If in the last equation there be put pi=o, then u'mean=o•708af (d/i'I); where the velocity is See also:

independent of the pressure po at the other end, a result which apparently must be absurd. Probably for long pipes, as for orifices, there is a limit to the ratio of the initial and terminal pressures for which the formula is applicable. X. FLOW IN See also:RIVERS AND CANALS § 95. Flow of Water in Open Canals and Rivers.—When water flows in a pipe the section at any point is determined by the form of the boundary. When it flows in an open channel with free upper surface, the section depends on the velocity due to the dynamical conditions. Suppose water admitted to an unfilled See also:canal. The channel will gradually fill, the section and velocity at each point gradually changing. But if the inflow to the canal at its head is constant, the increase of cross section and diminution of velocity at each point attain after a time a limit. Thenceforward the section and velocity at each point are constant, and the motion is steady, or permanent regime is established. If when the motion is steady the sections of the stream are all equal, the motion is uniform.

By See also:

hypothesis, the inflow 9v is constant for all sections, and Sl is constant; therefore v must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the See also:bed has a uniform slope. In that case the motion is uniform, the See also:depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section to section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a See also:river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections. In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams See also:rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a See also:plane layer abab (fig. 102) between sections normal From (1) and (3) But (14) (15) (16) to the direction of motion is treated as sliding down the channel to variation of the coefficient of friction with the velocity, proposed an a'a'b'b' without deformation. The component of the weight parallel expression of the form to the channel bed balances the friction against the channel, and 3 =a(1+i3/v), (5) in estimating the friction the velocity of rubbing is taken to be the and from 255 experiments obtained for the constants the values mean velocity of the stream.

In actual streams, however, the a=0.007409; /3=0.1920. velocity of rubbing on which the friction depends is not the mean ~ This gives the following values at different velocities: 11= 0.3 0.5 0.7 I 11 2 3 5 7 10 15 0.01215 0.01025 0.00944 o.o0883 0.00836 0.00812 0.90788 0.00769 0.00761 0.00755 0.00750 velocity of the stream, and is not in any simple relation with it, for channels of different forms. The theory is therefore obviously based on an imperfect hypothesis.

End of Article: Q00 0

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