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Encyclopedia Britannica - Main :: DIO-DRO |
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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,2I. (5a) Thus if ci, =o.97, then cr=oo628, That is, for such an orifice about 61% of the head is expended in overcoming frictional resistances to flow. Coefficient of ContractionSharp-edged Orifices in Plane Surfaces.When a jet issues from an aperture in a vessel, it may either 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 condition will be found if the orifice is bevelled outwards as at a, so as to be sharp
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 con-verging towards the orifice continue to converge beyond it, so that the 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 change of direction of motion at the edge of the orifice, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let w be the 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 kind of orifice, called the coefficient of contraction. When the orifice is a sharp
Coefficient of Discharge.In applying the general formula Q=wv to a stream, it is assumed that the filaments have a 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= cccwJ (2gh), or simply, if c=ccc, Qa = cwsl (2gh), where c is called the coefficient of discharge. Thus for a sharp-edged plane orifice c=097X 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 fine pitch (fig. 16) are convenient. These are set to 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 horizontal jet. Let OX, OY (fig. 17) be horizontal and 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 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 case of large orifices such as weirs, the velocity can be directly determined by using a Pitot tube ( 144). The coefficient of discharge, which for 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 simple in principle are not 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 wall
into the waste channel. D is a stilling screen to prevent agitation of the surface at the measuring point, E, and F is a discharge 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 water surface in the supply reservoir, the coefficients under different conditions of head are pretty exactly known. Suppose the same quantity of water is made to flow in succession through such an orifice and through another orifice of which the coefficient is required, and when the rate of flow is constant the heads over each orifice are noted. Let hi, h2 be the heads, wi, 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 mouthpiece exactly of the form of the D=1asd liil~ ij6i91 iI Eliminating t, Then e,..._ ~: II ~,~: k 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 account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York
Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged, with free Discharge into the Air. Q=cwJ (2gh). Head Diameters of Orifice. measured to 02 04 10 20 40 60 1.0 Centre of Orifice. Values of C. 0.3 .. 6z1 0.4 .637 618 . o6 655 630 613 6o1 596 588 0.8 648 626 610 6o1 597 '594 '583 1o .644 .623 '6o8 .60o -598 '595 '591 2o .632 614 604 599 '599 .597 .595 4 0 623 .609 602 599 598 '597 '596 8o .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 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- Long Iong 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 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 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 top of next 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. 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 1o 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 (Treatise on Water Supply 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 interpolation , all the most reliable experiments being plotted and curves drawn
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 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 1o .. 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 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 internal rim or border is applied to 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 o66 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 o666 0.710 0.694 0.696 .656 0.609 0.653 0.657 0.640 0.672 0.675 0.685 0.687 o688 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 o656 o660 0.641 0.675 0.678 0.683 0.693 0.695 0.692 0.709 0.711 1.968 o618 0.649 0.653 0.640 0.676 0.679 o678 0.695 0.697 o688 0.710 0.712 3.28 o6o8 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 o618 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 0J92 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 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 oI64 487 .571 .6o6 617 626 628 .627 .623 618 ) C '585 614 .633 .645 .652 651 650 .65o '649 25. 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-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 influence of the sides begins to be 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. ( 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 series of singular changes of form after leaving the orifice. These were first investigated by G. Bidone (17811839); subsequently H. G. Magnus (18021870) measured jets from different orifices; and later Lord Rayleigh (Proc. Roy. Soc. 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 regular 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 meeting streams. If two equal streams having the same 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 sheet perpendicular to the plane of the streams. Let al, 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 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. Buff (18051878), namely, that it is due to surface tension. 26. Influence of Temperature on Discharge of Orifices.Professor W. C. Unwin found (Phil. Mag., 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 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 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 oo828 and 0.0391, showing that the resistance decreases to about half at the higher temperature. 27. Fire Hose Nozzles. Experiments have been made by J. R. Freeman on the coefficient of discharge from smooth 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 taper play-pipe, and the coefficient includes the resistance of this pipe (Amer. Soc. Civ. Eng. xxi.. 1889). Other forms of nozzle were tried such as ring nozzles for which the coefficient was smaller. IV. THEORY OF THE STEADY MOTION OF FLUIDS. 28. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed. (a) If the motion is rectilinear and uniform, the variation of pressure is the same as in a fluid at 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 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 independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any 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 side of the section, the distribution of pressure is the same as in a fluid at rest. End of Article: DISCHARGE OF If you wish, you can link directly to this article.
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