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Originally appearing in Volume V14, Page 64 of the 1911 Encyclopedia Britannica.
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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 the results obtained with the disks and Froude's results on planks 5o ft. long. The values given are the resistances per square foot at io t. per sec. Froude's Experiments. Tinfoil surface . 0.232 Varnish 0.226 Fine sand 0.337 Medium sand . . 0.456 Disk Experiments. 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 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 (Prot. 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 to, 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 in is an element of area on the disk moving with the velocity v, the friction on this element is fwvn, where f and n are constant for any given kind of surface. Let a 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 art-de, its velocity ar and the friction of this ring is f2lrrdra"rn. The moment of the friction about the axis of rotation is 2,ra"frn+'-dr, and the total moment of friction for the two sides of the disk is M = 4.iranfjo rn+2dr = (4aan/(n+3) If Rn+' If N is the number of revolutions per sec., M = (2 n-Yl.~n+l Nn / (ry .~.. 3) If Rn+3 and the work expended in rotating the disk is Ma=12n+3trn+2Nn+l/(n+3)}fRn+3 foot lb per sec. The experiments give directly the values of M for the disks corre- Part of the water in contact with the board at any point, and receiv- is the surface friction o the pipe, an it is convenient to estimate ing energy of motion from it, passes afterwards to distant regions of this independently of some smaller resistances which will be ac still water, and portions of still water are fed in towards the board counted 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 there- fore the work of the external forces and of the surface friction must be equal. Let fig. 8o 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 lections being ----is p and p+dp lb per square foot. Further, FIG. 80. let Q be the volume of flow or discharge of the pipe per second, S2 the area of a normal cross section, and x the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ lb, and fall through a height-dz ft. The work done by gravity is then -GQdz; 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 lb per square foot of the cross section. The work of this pressure on the volume Q is -Qdp. The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is x dl. The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3) - 3 Gxdlv3/2g, or, since Q = Slv, -I'G(x/S2)Q(v2 2g)dl; x 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,- - GQdz — Qdp — I G (x/ft)Q (v2/2g)dl = o. Dividing by GQ, dz+dp/G+l'(x/t2) (v2/2g)dl =o. Integrating, z+p/G+r(x cl)(v2/2g)l=constant. (I) § 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values pi, z1, li, and p2, z2, 12, respectively. Then zi +pi/G + 1-(x/f2) (v2/2g)li = z2+p2/G +i (x/[1) (v2/2g)l2 ; or, if l2 =L, }rearrangi /ng the terms, / ]v2/2g=(I/L)[ (zi+pi/G)-(z2+P2/G)}f2/x• (2) Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights pi/G and p2/G due to the pressures pi and p22 at A and B. Hence (zi+pi/G)—(z2+p2/G) is the quantity represented in the figure by DE, the fall of level of the pressure columns, or virtual fall of 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 ft/x which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m. Introducing these values, 1' v2/2g = mh/L = mi. For pipes of circular section, and diameter d, m= x ='-s2rd'-/2rd = id. i'v2/2g= 4dh/L =-,di; h =1'(4L/d) (v2/2g) ; which shows that the head lost in friction is proportional to the head due to the velocity, and is found by multiplying that head by the coefficient 41L/d. It is assumed above that the atmospheric pressure at C and D is the same, and this is usually nearly the case. But if C and D are at greatly different levels the excess of barometric pressure at C, in feet of water, must be added to p2/G. § 73. Hydraulic Gradient or Line of Virtual Slope.—Join CD. Since the head lost in friction is proportional to L, any intermediate pressure column between A and B will have its free surface on the line CD, and the vertical distance between CD and the pipe at any point measures the pressure, exclusive of atmospheric pressure, in the pipe at that point. If the pipe were laid along the line CD instead of AB, the water would flow at the same velocity by gravity without any change of pressure from section to section. Hence CD is termed the virtual slope or hydraulic gradient of the pipe. It is the line of free surface level for each point of the pipe. If an ordinary pipe, connecting reservoirs open to the air, rises at any joint above the line of virtual slope, the pressure at that point is less than the atmospheric pressure transmitted through the pipe. At such a point there is a liability that air may be disengaged from the water, and the flow stopped or impeded by the accumulation of air. If the pipe rises more than 34 ft. above the line of virtual slope, the pressure is negative. But as this is impossible, the continuity of the flow will be broken. If the pipe is not straight, the line of virtual slope becomes a curved line, but since in actual pipes the vertical alterations of level are generally small, compared with the length of the pipe, distances measured along the pipe are sensibly proportional to distancesmeasured along _the - -hori-L-zontal ---- e- Vi2 projrtectil ua-on sloofp .the pipe. Hence the line of hydraulic gradient may be taken to be a straight line without error of practical importance. • § 74. Case of a Uniform Pipe connecting two Reservoirs, when all the Resistances are taken into account.—Let h (fig. 82) be the difference of level of the reservoirs, and v the velocity, in a pipe of length L and diameter d. The whole work done per second is virtually the removal of Q cub. ft. of water from the surface of the upper reservoir to the surface of the lower reservoir, that is GQh foot-pounds. This is expended in three ways. (I) The head v2/2g, corresponding to an expenditure of GQv2/2g foot-pounds of work, is employed in giving energy of motion to the water. This is ulti- V -'rz Ti -- mately wasted in eddying motions in the lower reservoir. (2) A portion of head, which experience shows may be expressed in the form 1'ov2/2g, corresponding to an expenditure of GQov2/2g foot-pounds of work, is employed in overcoming the resistance at the entrance to the pipe. (3) As already shown the head expended in overcoming the surface friction of the pipe is 1'(4L/d) (v2/2g) corresponding to GQg(4L/d)(v2/2g) foot-pounds of work. Hence GQh = GQv2/2g+GQI'ov2/2g+GQi-.4L.v2/d.2g ; h= (I +lo+1'. 4L/d)v2/2g. v = 8•o25~ [hd/{ (I +?-o)d+41-1.11. If the pipe is bellmouthed, io is about =•o8. If the entrance to the pipe is cylindrical, ho=0.505. Hence I+ro=I•o8 to 1.505. In general this is so small compared with 1-4L/d that, for practical calculations, it may be neglected; that is, the losses of head other than the loss in surface friction are left out of the reckoning. It is only in short pipes and at high velocities that it is necessary to take account of the first two terms in the bracket, as well as the third. For instance, in pipes for the supply of turbines, v is usually limited to 2 ft. per second, and the pipe is bellmouthed. Then 1.o8v2/2g =0.067 ft. In pipes for towns' supply v may rasige from 2 to 42 ft. per second, and then I.5v2/2g=o.1 to 0.5 ft. In either case this amount of head is small compared with the whole virtual fall in the cases which most commonly occur. When d and v or d and h are given, the equations above are solved quite simply. When v and h are given and d is required, it is better to proceed by approximation. Find an approximate value of d by assuming a probable value for 1• as mentioned below. Then from that value of d find a corrected value for (-and repeat the calculation. The equation above may be put in the form h=(41'/d)[{(I+o)d/41'}+L]v2/2g; (6) from which it is clear that the head expended at the mouthpiece is equivalent to that of a length (I+ro)d/41" of the pipe. Putting 1+1"0=1.505 and i'=o•o1, the length of pipe equivalent to the mouthpiece is 37.6 d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance in approximate calculations. § 75. Coefficient of Friction for Pipes discharging Water.—From the average of a large number of experiments, the value of 1- for ordinary iron pipes is =0.007567. (7) But practical experience shows that no single value can be taken applicable to very different cases. The earlier hydraulicians occupied themselves chiefly with the dependence of i- on the velocity. 1-laving regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that might be expressed in the form i- =a+P.'. The following are the best numerical values obtained for so ex-pressed : a R. de Prony (from 51 experiments) 0.006836 O.00I 1 16 J. F. d'Aubuisson de Voisins . 0.00673 0•00121I . . 0.005493 0.00143 J. A. Eytelwein Weisbach proposed the formula 41' = a+13/f v = 0.003598 +0.004289/ v. (8) C It - - ~ i ---i Datum Line Then or (3) (4) (4a) (5) § 76. Darcy's Experiments on Friction in Pipes.-All previous experiments on the resistance of pipes were superseded by the remarkable researches carried out by H. P. G. Darcy (18o3-1858), the Inspector-General of the Paris water works. His experiments were carried out on a scale, under a variation of conditions, and with a degree of accuracy which leaves little to be desired, and the results obtained are of very great practical importance. These results may be stated thus: I. For new and clean pipes the friction varies considerably with the nature and polish of the surface of the pipe. For clean cast iron it is about 11 times as great as for cast iron covered with pitch. 2. The nature of the surface has less influence when the pipes are old and incrusted with deposits, due to the action of the water. Thus old and incrusted pipes give twice as great a frictional resistance as new and clean pipes. Darcy's coefficients were chiefly determined from experiments on new pipes. He doubles these coefficients for old and incrusted pipes, in accordance with the results of a very limited number of experiments on pipes containing incrustations and deposits. 3. The coefficient of friction may be expressed in the form r=a+fl/v; but in pipes which have been some time in use it is sufficiently accurate to take 3-=at simply, where al depends on the diameter of the pipe alone, but a and on the other hand depend both on the diameter of the pipe and the nature of its surface. The following are the values of the constants. For pipes which have been some time in use, neglecting the term depending on the velocity; = a(I +R/d). a For drawn wrought-iron or smooth cast- (9) These coefficients may be put in the following very simple form, iron pipes . . . . .004973 .084 For pipes altered by light incrustations . .00996 •084 without sensibly altering their value: For clean pipes i-=.005(1+1/12d) (9a) For slightly incrusted pipes . = .01(1 +1/12d) Darcy's Value of the Coefficient of Friction i- for Velocities not less than 4 in. per second. Diameter ' Diameter of Pipe of Pipe in Inches. in Inches. New Incrusted pipes. Pipes. New Incrusted pipes. Pipes. 2 0.00750 0.01500 18 .00528 .01056 3 .00667 •01333 21 •00524 •01048 4 •00625 .01250 24 •0052I .01042 5 •oo600 •01200 27 •00519 •01037 6 •00583 .01167 30 .00517 •01033 7 •00571 .01143 36 •00514 .01028 8 .00563 •01125 42 •00512 •01024 9 •00556 •01111 48 .00510 •01021 12 .00542 .01083 54 •00509 .01019 L 15 .00533 .01067 These values of ;• are, however, not exact for widely differing velocities. To embrace all cases Darcy proposed the expression =(a+add)+($+Oi/d2)/v; (Io) which is a modification of Coulomb's, including terms expressing the influence of the diameter and of the velocity. For clean pipes Darcy found these values a •004346 a, •0003992 14 = •ooto182 p1= •000005205. It has become not uncommon to calculate the discharge of pipes by the formula of E. Ganguillet and W. R. Kutter, which will be discussed under the head of channels. For the value of c in the relation v=c~ (mi), Ganguillet and Kutter take 41.6-+.81 I/n+•00281/i c= even selected experiments the values of the empirical coefficient range from 0.16 to o•oo28 in different cases. Hence means of dis- criminating the probable value of 3• are necessary in using the equations for practical purposes. To a certain extent the knowledge that decreases with the size of the pipe and increases very much with the roughness of its surface is a guide, and Darcy's method of dealing with these causes of variation is very helpful. But a further difficulty arises from the discordance of the results of different experiments. For instance F. P. Stearns and J. M. Gale both experimented on clean asphalted cast-iron pipes, 4 ft. in diameter. Ac-cording to one set of gaugings •0051, and according to the other ;•_ .0031. It is impossible in such cases not to suspect some error in the observations or some difference in the condition of the pipes not noticed by the observers. It is not likely that any formula can be found which will give exactly the discharge of any given pipe. For one of the chief factors in any such formula must express the exact roughness of the pipe surface, and there is no scientific measure of roughness. The most that can be done is to limit the choice of the coefficient for a pipe within certain comparatively narrow limits. The experiments on fluid friction show that the power of the velocity to which the resistance is proportional is not exactly the square. Also in deter-mining the form of his equation for Darcy used only eight out of his seventeen series of experiments, and there is reason to think that some of these were exceptional. Barre de Saint-Venant was the first to propose a formula with two constants, dh/4l = mV", where m and is are experimental constants. If this is written in the form log m+n log v=log (dh/4l), we have, as Saint-Venant pointed out, the equation to a straight line, of which m is the ordinate at the origin and n the ratio of the slope. If a series of experimental values are plotted logarithmically the determination of the constants is reduced to finding the straight line which most nearly passes through the plotted points. Saint-Venant found for n the value of 1.71. In a memoir on the influence of temperature on the movement of water in pipes (Berlin, 1854) by G. H. L. Hagen (1797-1884) another modification of the Saint-Venant formula was given. This is h/l=mv"/d=, which involves three experimental coefficients. Hagen found n=1.75; x=1.25; and m was then nearly independent of variations of v and d. But the range of cases examined was small. In a remarkable paper in the Trans. Roy. Soc., 1883, Professor Osborne Reynolds made much clearer the change from regular stream line motion at low velocities to the eddying motion, which occurs in almost all the cases with which the engineer has to deal. Partly by reasoning, partly by induction from the form of logarithmically plotted curves of experimental results, he arrived at the general equation h/l=c(v'/d3-")P2-", where n =1 for low velocities and n=1.7 to 2 for ordinary velocities. P is a function of the temperature. Neglecting variations of temperature Reynold's formula is identical with Hagen's if x=3-n. For practical purposes Hagen's form is the more convenient. Values of Index of Velocity. Surface of Pipe. Authority. Diameter Values of n. of Pipe in Metres. Tin plate Bossut •036 1.697 1 2 7 '054 I•730 Wrought iron (gas Hamilton Smith -i 1'75 .026 pipe) .014 I.77o 1.866l Lead . . Darcy .027 1.755 r 1.77 L .041 I .76o .1 Clean brass Mair •036 1.795 I.795 Hamilton Smith •0266 1.7601 Lampe . 4185 I .85o Asphalted W. W. Bonn .306 j- p85 I.582 Stearns 1.219 1.88o Riveted wrought .2776 P804 iron } Hamilton Smith .3219 1.892 P87 3749 I1'85.892 2 Wrought iron (gas • I.90 o pipe) Darcy . . .o266 1.894 1.87 '0395 I.838 .•0819 1.950 New cast iron . Darcy • i88 I ,957 195 •50 1.950 '0364 I.835 Cleaned cast iron Darcy . . •080I 2.000 .2447 2.000 2.00 397 2.07 .0795 Incrusted cast iron Darcy 1.990 1.990 2.00 jL .0795 1'990 '2432 1+ [(41.6 + .00281/a)(nN m)] where n is a coefficient depending only on the roughness of the pipe. For pipes uncoated as ordinarily laid n=0.013. The formula is very cumbrous, its form is not rationally justifiable and it is not at all clear that it gives more accurate values of the discharge than simpler formulae. § 77. Later Investigations on Flow in Pipes.-The foregoing statement gives the theory of flow in pipes so far as it can be put in a simple rational form. But the conditions of flow are really more complicated than can be expressed in any rational form. Taking ^^~HHII^^^I^ ^^^ UI1II^ I ^^IIHII^ ^^^ E Ni NM El ME •a ^^ ^^^^^^^^^~ ^^^^^^^^^^^^^^^^ ^^^^iii^^^^^^^^^^^ ^^^^^ ^^^ ^^^^^^^~~^ '3 ^^^^^^^^^^ ^^. 0.52 _ ,. ^^^ ^^^^:~~^ ^^^ E ri E 2 ^^^^^^^ ^^^ ^ gl°fie tp fi s,._i~{P~~t ^^^^_lp ~.^^ St ^^^^^^^^^^^^~~ a ~'~ / ,^^ ^^^^^^ ^ ^ !IEEE Ea StO, FAME ^-'~• !PAM ^ ••IR^^^^^^^^^^^_^^^ ^^^^^^^^ ^^^^^ ^^^^^^ ^^^^^^^^^^^^ ~~ , e~ P ^^^^ ®= Mt 1+°'~ , ^^ ^^^^ l ^^^^^^^^^^^^^^^^^^^^^^^~ 'fi ^~^^ ^a^^ NEI o a!, ^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^aa..,s PA fi. s ^^ ^ ^ ' ^ ^ ^^^^^^ ^ ^^^^^^^^^^^^^^^ )Mg u..R• ^ d i~ ,P'4> t. ^^^^^^^^^^^^^ .7 ^^ ^^^^^^^^^^^^ t° pIC ~i^^~F ^^^^^^^^^^ ^^^,o.N^Il7'a Mrs^ ,U^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^ ^^ 6 ^^^^^^^^^RU^ ^^~if g95 ^ ^.^ ^..^^^^^ ..^.^ x.838 .~^..M~... ^^^^^^^ S\;ems''', f/ ^^^ii^^^^^^^^^^^^^^^^^^^^^^^ ,svve ^^^^^^^ ^^^^^^^^^^^ ^^^^^ ^^HI1HI^^^^^ •1.1 ^^^^ ^ ^ ^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^ ^^^^^^^^^^^^• 111I . ^^^^^^^^^^^^^^^^. ^^^^^^^ 2.9 1.0 .1 •2 •3 .4 •5 •9 .7 -8 .9 0.0 •1 ' •2 •a .4 In 1886, Professor W. C. Unwin plotted logarithmically all the most trustworthy experiments on flow in pipes then available.' Fig. 83 gives one such plotting. The results of measuring the slopes of the lines drawn through the plotted points are given in the table. It will he seen that the values of the index n range from 1.72 for the smoothest and cleanest surface, to 2.00 for the roughest. The numbers after the brackets are rounded off numbers. The value of n having been thus determined, values of mid' were next found and averaged for each pipe. These were again plotted logarithmically in order to find a value for x. The lines were not very regular, but in all cases the slope was greater than I to 1, so that the value of x must be greater than unity. The following table gives the results and a comparison of the value of x and Reynolds's value 3-n. Kind of Pipe. n 3-n x Tin plate . 1.72 2.28 I.10o Wrought iron (Smith) '•75 1.25 1.2I0 Asphalted pipes I.85 1.15 1.127 Wrought iron (Darcy) 1.87 1.13 1.68o Riveted wrought iron P87 1.13 '.390 New cast iron . 1.05 1.168 Cleaned cast iron 2. 00 1.00 1.168 Incrusted cast iron . 2.00 1.00 1.16o With the exception of the anomalous values for Darcy's wrought- iron pipes, there is no great discrepancy between the values of x and 3-n, but there is no appearance of relation in the two quantities. For the present it appears preferable to assume that x is independent of n. It is now possible to obtain values of the third constant m, using the values found for n and x. The following table gives the results, the values of m being for metric measures. " Formulae for the Flow of Water in Pipes," Industries (Man- chester, 1886).Here, considering the great range of diameters and velocities in the experiments, the constancy of in is very satisfactorily close. The asphalted pipes give lather variable values. But, as some of these were new and some old, the variation is, perhaps, not surprising. The incrusted pipes give a value of m quite double that for new pipes but that is perfectly consistent with what is known of fluid friction in other cases. Kind of Pipe. Diameter Value of Mean Authority. in value Metres. m. of m. Tin '0' } •01686 Bossut 696 plate {0,054 Wrought iron {o oz7 0139} '01310 Hamilton Smith 0.027 •01749 Hamilton Smith o•3o6 .02058 W. W. Bonn Asphalted 0.306 •02107 W. W. Bonn pipes 0.419 .01650 .01831 Lampe 1.219 .01317 Stearns 1.219 •02107 Gale 0.278 .01370 Riveted 0'322 .0'440 wrought iron 0'375 .01390 '01403 Hamilton Smith 0.432 .01368 0.657 .01448 0.082 .017251 New cast iron JO'137 ` 1658 Darcy '0 .01734 I 734 0 500 .017451 Cleaned cast 0.080 01979 iron 0.245 .02091 •01994 Darcy o.297 .01913 Incrusted cast 0'036 03693 iron 0.080 .03530 •03643 Darcy 0.243 .03706 General Mean Values of Constants. The general formula (Hagen's)-h,/l=my"/d'.2g-can therefore be taken to fit the results with convenient closeness, if the following mean values of the coefficients are taken, the unit being a metre:- Kind of Pipe. in x n , Tin plate .0169 1.10 1.72 Wrought iron .0131 I.21 1.75 Asphalted iron . .0183 1.127 1.85 Riveted wrought iron .0140 1.390 1.87 New cast iron . •0166 1.168 1.95 Cleaned cast iron .0199 1.168 2.0 Incrusted cast iron •0364 1.160 2.0 L The variation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known. It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:- Kind of Pipe. m x n Tin plate .0265 1.10 1.72 Wrought iron .0226 1.21 1.75 Asphalted iron . .0254 1.127 I.85 Riveted wrought iron .026o 1.390 1.87 New cast iron . . .0215 1.168 1.95 Cleaned cast iron .0243 1.168 2.0 Incrusted cast iron •0440 I.16o 2.0 § 78. Distribution of Velocity in the Cross Section of a Pipe.-Darcy made experiments with a Pitot tube in 1850 on the velocity at different points in the cross section of a pipe. He deduced the relation V -v =11.3(r3/R),/ where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (Mem. de l'Academie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b= Ri/U2, where U is the mean velocity in the pipe; then V/U =1+9.03.,/ b. A very useful result for practical purposes is that at 0.74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin. § 79. Influence of Temperature on the Flow through Pipes.-Very careful experiments on the flow through a pipe 0.1236 ft. in diameter This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynolds's equation is assumed h =nzLV' /d3-", and n is taken 1.795, then values of m at each temperature are practically constant- Temp. F. m. Temp. F. in. 57 0.000276 loo 0.000244 70 0.000263 110 0.000235 8o 0.000257 120 0.000229 90 0.000250 130 0.000225 160 0.000206 where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different temperatures Professor W. C. Unwin found that the resistance was proportional to constant X (I-o•oo21t) and the values of m given above are expressed almost exactly by the relation M=0'000311(1-0'00215 t). In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3 % of resistance for 10° F. increase of temperature. § 80. Influence of Deposits in Pipes on the Discharge. Scraping Water Mains.-The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. Jn pipes which convey certain kinds of water, oxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1858, 14 M. in length, consists of 10-in., 9-in. and 8-in. pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to 51 % of the original discharge. J. G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see " Incrustation of Iron Pipes," by W. Ingham, Proc. Inst. Mech. Eng., 1899). It was found that by scraping the interior of the pipe the discharge was increased 56 %. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly to To % and afterwards more slowly, the diminution in a year being about 25%. Fig. 85 shows a scraper for water mains, similar to Appold's but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or with-drawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the \V\0.\\\\Q~\Ul\l\~O\`:.\\1\`1\\\Ciil`V\\VC\\RO\1W0\gV\\\\UOVVti+\. ~p~\4t~l~Vl\l\G and 25 ft. long, with water at different temperatures, have been made by J. G. Mair (Proc. Inst. Civ. Eng. Ixxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 14 in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v''"5, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h=i'(4L/D)(v2/2g), then for heads varying from 1 ft. to nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft. per second, the values of were as follows:- Temp. F. Temp. F. g' 57 .0044 to •0052 100 .0039 to •0042 70 .0042 t0 •0045 II0 •0037 t0 •0041 80 •0041 to .0045 120 •0037 t0 •0041 90 .0040 to •0045 130 •0035 to .0039 16o .0035 to `0038FIG. 85. Scale -;. main. At Lancaster after twice scraping the discharge was increased 564%, at Oswestry 541%. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation. The scraper can be easily followed when the mains are about 3 ft. deep by the noise it makes. y The average speed of the scraper at Torquay is 22 M. per hour. At Torquay 49 % of the deposit is iron rust, the rest being silica, lime and organic matter. In the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another character which has led to trouble in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to be an organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see " Deposits in Pipes," by Professor J. C. Campbell Brown, Eroc. Inst. Civ. Eng., 1903-1904). § 81. Flow of Water through Fire Hose.-The hose pipes used for fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by J. R. Freeman' (Am. Soc. Civ. Eng. xxi., 1889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. In reducing the results the true diameter has been taken. Let v=mean velocity in ft. per sec.; r=hydraulic mean radius or one-fourth the diameter in feet; i= hydraulic gradient. Then v=n' (ri). Diameter Gallons i v It in (United Inches. States) per min. Solid rubber SS 2.65 215 .1863 12.50 123.3 hose i 344 .4714 20.00 124.0 Woven cotton, 2.47 200 .2464 13.4o 119.1 rubber lined 299 •5269 20.00 121.5 Woven cotton, ss 2.49 200 .2427 13.20 117.7 rubber lined 2 319 .5708 21.00 122•I K nit cotton, { 2.68 132 .o8o9 7.50 111.6 rubber lined 299 '3931 17'00 114'8 i K nit cotton, s 2.69 204 .2357 II.50 I00•I rubber lined 319 .5165 18•oo Io5.8 ' Woven cotton, 2.12 154 '3448 1,4.00 113'4 rubber lined 2 24o .7673 21.81 118.4 Woven cotton, 2.53 54.8 •0261 3.50 94.3 rubber lined 2 298 .8264 19.00 91.0 Unlined linen 2.6o 57.9 '0414 3'50 73'9 hose 331 1.1624 20.00 79.6 § 82. Reduction of a Long Pipe of Varying Diameter to an Equivalent Pipe of Uniform Diameter. Dupuit's Equation.-Water mains for the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calculations may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main. H---- E 1 , yd In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let 1,, 12... be the lengths, dl, d2... the diameters, v,, v2... the velocities, i1, i2... the slopes, for the successive portions, and let 1, d, v and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is h=i111+i212-}-, (v,2.411/2gd,) -}- '(v22.412/2gd2) + .. . and in the uniform main it = '(v2.4l/2gd). If the mains are equivalent, as defined above, i-(v2.4112gd) = (v12.411/2gd1) +'(v22.412/2gd2) + .. . But, since the discharge is the same for all portions, '-;,rd2v= 4~d12v,= 1ird22v2=... vi = vd'/d,2 ; v2 = vd2/d22 .. . Also suppose that f may be treated as constant for all the pipes. Then l/d = (d4/d1") (l1/d,) -i- (d4/d2') (12/d2) -F•... 1 = (d5/d1')l14-(ds/d25)l2-l-... which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge. § 83. Other Losses of Head in Pipes.-Most of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted. Sudden Enlargement of Section.-Suppose a pipe enlarges in section from an area ceo to an area col (fig. 87) ; then vl/vo = wo/wl ; or, if the section is circular, vi/vo= (do/di)'. d., ,;);24 The head lost at the abrupt change o of velocity has already been shown to be the head due to the relative velocity of the two parts of the stream. Hence head lost %e = (vo -vi)2/2g = (wi/wo - I )2v12/2g = 1(d,/do)2 - 12v12/2g or I),=~ev12/2g, (I) if 1e is put for the expression in brackets. - 1.1 1.2 1.5 1.7 1.8 1.9 2.0 2.5 3.0 3.5 4.0 5.o 6.o 7.0 8.o COI/WO = dl/do = 1.05 1.10 1.22 1.30 1.34 1.38 1.41 1.58 1.73 1.87 2.00 2.24 2.45 2.65 2.83 = .01 .04 .25 .49 .64 Si .1.00 2.25 4.00 6.25 9.0016.0025.00 36.049.0 Abrupt Contraction of Section.-When water passes from a larger to a smaller section, as in figs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. --- .11 3 FIG. 88. Let w be the section and v the velocity of the stream at bb. At as the section will be cow, and the velocity (w/cow)v=v/cl, where co is the coefficient of contraction. Then the head lost is = (v/c,- v)2/2g = (1/co - I )2v2/2g ; and, if co is taken 0.64, IIm = 0.316 v2/2g. (2) The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by friction. For water entering a cylindrical, not bell-mouthed, pipe from a reservoir of indefinitely large size, experiment gives 0.505 v2/2g. (3) If there is a diaphragm at the mouth of the pipe as in fig. 89, let on be the area of this orifice. Then the area of the contracted stream is cowl, and the head lost is lc = 1(w/ecw,) - 112v2/2g = i-ov2/2g if r, is put for { (w/cowl) -112. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice :- wl/w = 0.1 0.2 0.3 0.4 0.5 o.6 0.7 o.8 0.9 1.0 co = .6,6 .614 .612 .6,o .617 .6o5 .603 .601 .5g8 .596 1•n = 231.7 50.99 19.78 9.612 5.256 3.077 1.876 1 169 0.734 0.480 When a diaphragm was placed in a tube of uniform section (fig. 90) I I the following values were obtained, col. being the area of the orifice and w that of the pipe:-

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