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See also:DISCHARGE OF LIQUIDS] The relation between c, and cr for any orifice is easily found: v„= cvJ2gh = J 12ygh/(1 +cr)] cv=J {I/(1-Fcr)). (5) cr= I/c,2—I. (5a) Thus if ci, =o.97, then cr=o•o628, That is, for such an orifice about 61% of the See also:head is expended in overcoming frictional resistances to flow. Coefficient of Contraction—Sharp-edged Orifices in See also:Plane Surfaces.—When a See also:jet issues from an See also:aperture in a See also:vessel, it may either See also:spring clear from the inner edge of the orifice as at a or b (fig. 15), or it may adhere to the sides of the orifice as at c. The former See also:condition will be found if the orifice is bevelled outwards as at a, so as to be See also:sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the See also:plate in which the aperture is formed is less than the See also:diameter of the jet. But if the thickness is greater the condition shown at c will occur. When the discharge occurs as at a or b, the filaments See also:con-verging towards the orifice continue to converge beyond it, so that the See also:section of the jet where the filaments have become parallel is smaller than the section of the orifice. The inertia of the filaments opposes sudden See also:change of direction of See also:motion at the edge of the orifice, and the convergence continues for a distance of about See also:half the diameter of the orifice beyond it. Let w be the See also:area of the orifice, and c,w the area of the jet at the point where convergence ceases; then cc is a coefficient to be determined experimentally for each See also:kind of orifice, called the coefficient of contraction. When the orifice is a sharp-edged orifice in a plane See also:surface, the value of c~ is on the See also:average 0.64, or the section of the jet is very nearly five-eighths of the area of the orifice. Coefficient of Discharge.—In applying the See also:general See also:formula Q=wv to a stream, it is assumed that the filaments have a See also:common velocity v normal to the section w. But if the jet contracts, it is at the contracted section of the jet that the direction of motion is normal to a transverse section of the jct. Hence the actual discharge when contraction occurs is Qa = c,,v X co= ccc„wJ (2gh), or simply, if c=c„cc, Qa = cwsl (2gh), where c is called the coefficient of discharge. Thus for a sharp-edged plane orifice c=0•97X 0.64 =0.62. § 18. Experimental Determination of c,,, c~, and c.—The co-efficient of contraction cc is directly determined by measuring the dimensions of the jet. For this purpose fixed screws of See also:fine See also:pitch (fig. 16) are convenient. These are set to See also:touch the jet, and then the distance between them can be measured at leisure. The coefficient of velocity is determined directly by measuring the parabolic path of a See also:horizontal jet. Let OX, OY (fig. 17) be horizontal and See also:vertical axes, the origin being at the orifice. Let h be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If va is the velocity at39 the orifice, and t the See also:time in which a particle moves from 0 to A, then x=vat ; Y=zgt2• va = J (gx2/2y). cr =va/J (2gh) =J (x2/4Yh). In the See also:case of large orifices such as weirs, the velocity can be directly determined by using a Pitot See also:tube (§ 144). The coefficient of discharge, which for See also:practical purposes is the most important of the three coefficients, is best determined by tank measurement of r the flow from the given orifice in a =_= T= suitable time. If —_ Q is the discharge A measured in the tank per second, then c=Q/wJ (2gh). Measurements of this kind though See also:simple in principle are not See also:free from some practical difficulties, and require much care. In fig. 18 is shown an arrangement of measuring tank. The orifice is fixed in the See also:wall of the cistern A and discharges either into the See also:waste channel BB, or into the measuring tank. There is a See also:short trough on rollers C which when run under the jet directs the discharge into the tank, and when run back again allows the discharge to drop into the waste channel. D is a stilling See also:screen to prevent agitation of the surface at the measuring point, E, and F is a discharge See also:valve for emptying the measuring tank. The rise of level in the tank, the time of the flow and the head over the orifice at that time must be exactly observed. For well made sharp-edged orifices, small relatively to the See also:water surface in the See also:supply See also:reservoir, the coefficients under different conditions of head are See also:pretty exactly known. Suppose the same quantity of water is made to flow in See also:succession through such an orifice and through another orifice of which the coefficient is required, and when the See also:rate of flow is See also:constant the heads over each orifice are noted. Let hi, h2 be the heads, wi, See also:w2 the areas of the orifices, c,, c2 the coefficients. Then since the flow through each orifice is the same Q =c,wis/ (2ghi) =c2w2J (2gh2). c2 = ct(0.7i/w2) J (hi/h2). § 19. Coefficients for Bellmouths and Bellmouthed Orifices.—If an orifice is furnished with a See also:mouthpiece exactly of the See also:form of the D=1•asd liil~ ij6i91 iI Eliminating t, Then e,..._ ~: II ~,~: k See also:list' A C at=0.8D. -"i contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, cr must be put =1. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is not done; and such a bellmouth may also have the form of the contracted jet. Fig. 19 shows the proportions of such a bellmouth or bellmouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose. For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.
Head over orifice, in ft. = h •66 1.64 11.48 55.77 337.93
Coefficient of velocity=c . '959 '967 '975 '994 '994
i Coefficient of resistance=cr •o87 •069 •052 •012 •012
As there is no contraction after the jet issues from the orifice, c=1, c=c,,; and therefore
Q =c,,(0\1 (2gh) =ceJ {2gh/(1 +cr)}.
§ 2o. Coefficients for Sharp-edged or virtually Sharp-edged Orifices.-There are a very large number of measurements of discharge from sharp-edged orifices under different conditions of head. An See also:account of these and a very careful tabulation of the average values of the coefficients will be found in the See also:Hydraulics of the See also:late See also: measured to •02 •04 •10 •20 •40 •60 1.0 Centre of Orifice. Values of C. 0.3 .. 6z1 0.4 .637 •618 . o•6 •655 •630 •613 •6o1 •596 •588 0.8 •648 •626 •610 •6o1 •597 '594 '583 1•o .644 .623 '6o8 .60o -598 '595 '591 2•o .632 •614 •604 •599 '599 .597 .595 4 0 •623 .609 •602 •599 •598 '597 '596 8•o .614 •605 •600 •598 .597 '596 '596 20.0 •601 '599 '596 '596 '596 '596 '594 At the same time it must be observed that See also:differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharp-edged orifice it is desirable that the coefficient for the particular orifice should be directly determined. The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University. Coefficient of Discharge for Sharp-edged Orifices. Form of Orifice. Square. Rectangular Ratio Rectangular Ratio of Sides 4:1. of Sides 16:r. Head in -• ft. Cir- Lon Tri- cular. Sides Dia- See also:Long I•ong Long Sides angular. vertical gonal Sides Sides Sides hori- vertical. vertical. horzontai-l. vertical. zontal. r •62o •627 •628 •642 .643 •663 •664 .636 2 •613 •62o '628 •634 •636 •65o .65t •628 4 •6o8 •616 •618 •628 .629 •641 •642 •623 6 .607 '614 •616 •626 •627 '637 '637 '620 8 •6o6 •613 .614 '623 .625 '634 '635 •619 10 •605 •612 •613 •622 •624 .632 .633 •618 12 •604 •6,1 •612 •622 •623 •631 •631 •618 14 -604 •610 •612 •621 •622 •630 •630 •618 16 .603 •610 •611 •620 •622 •630 .630 •617 18 .603 •610 •611 •62o •621 .630 •629 •616 20 •603 .609 •611 •620 •621 •629 •628 •616 The orifice was o. 196 sq. in. area and the reductions were made with g=32.176 the value for See also:Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases. It decreases a little as the See also:size of the orifice is greater. Very careful experiments by J. G. Mair (Prot. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on See also:top of next See also:column. The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. See also:Professor W. C. Unwin found that the results agree with the formula c = 0.607 5 -i- o • / J h - 0.003 7d, where h is in feet and d in inches.Coefficients of Discharge from Circular Orifices. Temperature 51 ° to 55°. 1 Head in Diameters of Orifices in Inches (d). feet h. I 14 11 I I4 2 24 2z 24 13 Coefficients (c). •75 •616 •614 •616 •6,0 •616 •612 •607 •607 •609 1•o •613 •612 •612 •611 •612 •611 •604 •6o8 •609 1.25 .613 •614 •610 •6o8 •612 •6o8 •6o5 .6o5 •6o6 1.50 •610 •612 •611 •6o6 •610 •607 •603 •607 •6o5 1.75 •612 •611 •611 •6o5 •611 .605 •604 •607 •6o5 2.00 •609 •613 •609 •6o6 •609 •6o6 .6o¢ .6o¢ •605 The following table, compiled by J. T. Fanning (See also:Treatise on Water Supply See also:Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the still-water surface at some distance from the orifice. The values were obtained by graphic See also:interpolation, all the most reliable experiments being plotted and curves See also:drawn so as to average the discrepancies. Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in Vertical Plane Surfaces. Head to Ratio of Height to Width. Centre of 4 2 } Orifice. 11 r i I ai ,a ° m a., yy ai C 'w A . m 440.0 ee ;O eo;C ai See also:mat m ai 4 4.1 .. -751. Feet. •3 . 3 .c 3 «•3 ~, - e N .. .... .... o- ow o .-t-1 w o.. 0.2 .. .. .. .. .. .. .. •6333 •3 •6293 •6334 '4 .. .. .. .. . 614o •6306 •6334 .5 .. .. .. .. •6050 •6150 .6313 •6333 6 .. .. '5984 •6063 •6156 •6317 •6332 '7 •• •• •. '5994 •6074 •6162. •6319 •6328 •8 .. .. •6130 •6000 •6082 .6165 .6322 •6326 •9 .. .. '6134 •6006 •6o86 •6168 .6323 '6324 1•o .. •6135 •6oio •6090 .6172 .6320 •6320 P25 . . •6188 •6140 •6o18 •6o95 •6173 .6317 '6312 1.50 .. •6187 •6144 •6026 •6100 •6172 •6313 •6303 1.75 .. •6186 •6145 •6033 •6103 •6168 •6307 •6296 2 .. •6183 •6144 •6036 .6104 •6166 •6302 •6291 2.25 •618o •6143 •6029 •6103 •6163 .6293 •6286 2.50 •6290 •6176 '6139 •6043 •6102 •6157 •6282 •6278 2'75 •6280 •6173 •6136 •6046 •6101 •6155 •6274 •6273 3 '6273 •6170 .6132 .6048 •6100 •6153 •6267 •6267 3.5 •6250 •616o •6123 •6050 '6094 •6146 .6254 •6254 4 '6245 •6150 •6110 .6047 •6o85 •6136 •6236 •6236 4'5 •6226 •6138 •6100 .6044 •6074 •6125 •6222 •6222 5 •6208 •6124 '6o88 •6038 •6063 •6114 •62o2 •6202 6 •6158 •6094 •6063 •6020 •6044 •6087 •6154 •6154 7 •6124 .6064 •6038 •6ot1 •6o32 •6058 .6110 •6114 8 .6090 .6036 •6o22 •60,0 •6022 •6033 .6073 •6o87 9 •6o6o •6020 •6014 •6oto •6015 •6020 .6045 •6070 to •6035 .6015 •6oto •60,0 •6o,o •6o,o •6030 •6o6o 15 •6o4o •6o,8 •60,0 •6oit •6012 •6013 •6033 •6o66 20 •6045 .6024 •6012 •6012 •6014 •6018 .6036 •6074 25 •6048 •6o28 •6014 •6012 •6o,6 •6022 •6040 '6083 30 •6054 .6034 •6017 •6013 '6o,8 .6027 •6044 •6o92 .35 •6o6o •6039 •6021 .6014 •6o22 •6o32 •6049 .6103 40 •6o66 .6045 '6025 •6015 •6026 '6037 .6055 •6114 45 •6o54 •6o52 •6029 .6o,6 •6o3o •6043 •6o62 •6125 50 •6o86 •6o6o .6034 •6o18 •6o35 .6050 .6070 •6140 § 21. Orifices with Edges of Sensible Thickness.-When the edges of the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangerrieats of the orifice shown in vertical section at P, Q, R (fig. 20). The See also:plan of all the orifices is shown at S. The planks. forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case, Q=cb(H-h)J {2g(H+h)/2}. § 22. Partially Suppressed Contraction.-Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an See also:internal rim or border is applied to See also:part of the edge of the orifice (fig. 21), the convergence for so much of the edge is suppressed. For such cases G. Bidone found the following empirical formulae applicable: Table of Coefficients of Discharge for Rectangular Vertical Orifices in Fig. 20. Head h Height of Orifice, H -h, in feet. above upper 1.31 o•66 0.16 0.10 edg - of Orifi e in feet. P Q R P Q R P Q R P Q R 0.328 0.598 0.644 0.648 0.634 0.665 0.668 0.691 0.664 o•666 0.710 0.694 0.696 .656 0.609 0.653 0.657 0.640 0.672 0.675 0.685 0.687 o•688 0.696 0.704 0.706 •787 0.612 0.655 0.659 0.641 0.674 0.677 0.684 0.690 0.692 0.694 0.706 0.708 .984 0.616 o•656 o•660 0.641 0.675 0.678 0.683 0.693 0.695 0.692 0.709 0.711 1.968 o•618 0.649 0.653 0.640 0.676 0.679 o•678 0.695 0.697 o•688 0.710 0.712 3.28 o•6o8 0.632 0.634 0'638 0'674 0.676 0.673 0.694 0.695 0.68o ti 0.704 0.705 4.27 0.602 0.624 0.626 0.637 0'673 0.675 I 0.672 0.693 0.694 0.678 0.701 0.702 4'92 0'598 l 0.620 0.622 0.637 I 0'673 0.674 0.672 0.692 0.693 0.676 0.699 0.699 5.58 0.596 o•618 0.62o 0.637 0.672 0.673 0.672 0.692 0.693 0.676 0.698 0.698 6.56 0.595 0.615 0.617 0.636 0.671 0.672 0.671 0.691 0.692 0.675 0.696 0.696 9.84 0•J92 o.611 0.612 0.634 0.669 0.670 0.668 0.689 0.690 0.672 0.693 0.693 For rectangular orifices, e~ = 0.62(1+0.152707) ; and for circular orifices, c , =0.62(1 +o• 128n/p) ; when n is the length of the edge of the orifice over which the border extends, and p is the whole length of edge or perimeter of the orifice. The following are the values of cc, when the border extends over 4 or ; of the whole perimeter:- c, Circular Orifices. Rectangular Orifices. 0'643 •64o 0.667 •66o 0.691 .68o For larger values of nip § 24. Orifices Furnished with Channels of Discharge.-These ex- ternal See also:borders to an orifice also modify the contraction. The following coefficients of discharge were obtained with openings 8 in. wide, and small in proportion to the channel of approach (fig. 22, A, B, C). h, in feet. k_,-f-eeth, in . I '0656 '164 '328 •656 3'28 4:92 6'56 9'84 A •48o 511 '542 '599 •6oi •60I •6oi •6oi B 0.656 •48o •510 '538 •506 .592 •600 •602 •602 •6oi C .527 '553 '574 .592 •607 •6io •6io •609 •6o8 A ( '48 '577 .624 .631 .625 •624 .619 •613 •6o6 B o•I64 487 .571 .6o6 •617 •626 •628 .627 .623 •618 ) C '585 •614 .633 .645 .652 •651 •650 .65o '649 § 25. See also:Inversion of the Jet.-When a jet issues from a horizontal orifice, or is of small size compared with the head, it presents no b n/ p 0.25 0.50 0.75 the formulae are not applicable. C. R. Bornemann has shown, however, that these formulae for suppressed con-See also:traction are not reliable. § 23. Imperfect Contraction.-If the sides of the vessel approach near to the edge of the orifice, they interfere with the convergence of the streams to which the contraction is due, and the contraction is then modified. It is generally stated that the See also:influence of the sides begins to be See also:felt if their distance from the edge of the orifice is less than 2-7 times the corresponding width of the orifice. The coefficients of contraction for this case j marked peculiarity of form. But if the orifice is in a vertical surare imperfectly known. ( See also:face, and if its dimensions are not small compared with the head, C Slope I in 20 1 A h, h, 10 B it undergoes a See also:series of singular changes of form after leaving the orifice. These were first investigated by G. Bidone (1781—1839); subsequently H. G. See also:Magnus (1802—1870) measured jets from different orifices; and later See also:Lord See also:Rayleigh (Proc. See also:Roy. See also:Soc. See also:xxix. 71) investigated them anew. Fig. 23 shows some forms, the upper figure giving the shape of the orifices, and the others sections of the jet. The jet first contracts as described above, in consequence of the convergence of the fluid streams within the vessel, retaining, however, a form similar to that of the orifice. Afterwards it expands into sheets in planes perpendicular to the sides of the orifice. Thus the jet from a triangular orifice expands into three sheets, in planes bisecting at right angles the three sides of the triangle. Generally a jet from an orifice, in the form of a See also:regular See also:polygon of n sides, forms n sheets in planes perpendicular to the sides of the polygon. Bidone explains this by reference to the simpler case of See also:meeting streams. If two equal streams having the same See also:axis, but moving in opposite directions, meet, they spread out into a thin disk normal to the common axis of the streams. If the directions of two streams intersect obliquely they spread into a symmetrical See also:sheet perpendicular to the plane of the streams. Let al, See also:a2 (fig. 24) be two points in an orifice at depths hr, h2 from the free surface. The filaments issuing at al, a2 will have the different velocities d 2ghi and if 2gh2. Consequently they will tend to describe parabolic paths click and ascbs of different horizontal range, and intersecting in the point c. But since two filaments cannot simultaneously flow through the same point, they must exercise mutual pressure, and will be deflected out of the paths they tend to describe. It is this mutual pressure which causes the expansion of the jet into sheets. Lord Rayleigh pointed out that, when the orifices are small and the head is not See also:great, the expansion of the sheets in directions perpendicular to the direction of flow reaches a limit. Sections taken at greater distance from the orifice show a contraction of the sheets until a compact form is reached similar to that at the first contraction. Beyond this point, if the jet retains its coherence, sheets are thrown out again, but in directions bisecting the angles between the previous sheets. Lord Rayleigh accepts an explanation of this con-traction first suggested by H. See also:Buff (1805—1878), namely, that it is due to surface tension. § 26. Influence of Temperature on Discharge of Orifices.—Professor W. C. Unwin found (Phil. Mag., See also:October 1878, p. 281) that for sharp-edged orifices temperature has a very small influence on the discharge. For an orifice i cm. in diameter with heads of about i to I ft. the coefficients were: Temperature F. . 205°° 6 2 For a conoidal or See also:bell-mouthed orifice i cm. diameter the effect of temperature was greater: Temperature F. C. 190° 0.987 130° - 0.974 ° 94 60 o• 2 an increase in velocity of discharge of 4% when the temperature increased 130°. J. G. Mair repeated these experiments on a much larger See also:scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharp-edged orifice 21 in. diameter, with a head of 1.75 ft., the coefficient was 0.604 at 57° and 0.607 at 179° F., a very small difference. With a conoidal orifice the coefficient was 0.961 at 55° and 0.981 at 170° F. The corresponding coefficients of resistance are o•o828 and 0.0391, showing that the resistance decreases to about half at the higher temperature. § 27. See also:Fire See also:Hose Nozzles. Experiments have been made by J. R. See also:Freeman on the coefficient of discharge from smooth See also:cone nozzles used for fire purposes. The coefficient was found to be 0.983 for 1-in. nozzle; 0.982 fore in.; 0.972 for 1 in.; 0.976 for 18 in.; and 0.971 for 11 in. The nozzles were fixed on a See also:taper See also:play-See also:pipe, and the coefficient includes the resistance of this pipe (Amer. Soc. Civ. Eng. xxi.. 1889). Other forms of nozzle were tried such as See also:ring nozzles for which the coefficient was smaller. IV. THEORY OF THE STEADY MOTION OF FLUIDS. § 28. The general See also:equation of the steady motion of a fluid given under See also:Hydrodynamics furnishes immediately three results as to the See also:distribution of pressure in a stream which may here be assumed. (a) If the motion is rectilinear and See also:uniform, the variation of pressure is the same as in a fluid at See also:rest. In a stream flowing in anopen channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the See also:depth below the free surface. (b) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest. (c) If the fluid molecules take precisely the accelerations which they would have if See also:independent and submitted only to the See also:external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any See also:cross section is uniform and equal to the atmospheric pressure. (d) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either See also:side of the section, the distribution of pressure is the same as in a fluid at rest. Additional information and CommentsThere are no comments yet for this article.
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