Strickland, T. P.

The University of Sydney: Sydney eScholarship Journals online

, . /I THE FLOW OF WATER THROUGH SHARP-EDGED ORIFICES. (A paper read bifore the Sydney UniVfrsity Enginem'ng Socidy, on June 13th, 1906. By T. P. STRICKLAND, B.E., l\LSc. A.A.I.E.E. I NTRODUCTORY. Early in 1898, during tve pl'eparation of appal'atus for other experiments, the author made some determinations of the co-efficient of discharge of two sharp-edged circular orifices, and in working up the results for publication it has been found that the variation with the head of the co-efficient of discharge can be represented by a simple formula which is also applicable to tqe results obtained by other experimenters. A study of all the results available throws some light on the manner in which the co-efficient varies with the size and shape of the orifice. In this paper it is proposed to give an account of . the author's experiments, and of the development of the formula, together with a general discussion of the subject of the flow of water through sharp-edged orifices. As this discussion covers ground not usually gi ven ~Il the ordinary text books on hydraulics, it is hoped that it will be useful to the student members, while t he formulre may be of value to graduates in practice, 01' at all events form a guide to futnre workers in the same field. A rather lengthy description of the apparatus is given in the hope that it will prove useful to any who may have facilities fOl' continuing an investigation on the same subject which, though well-worn, still affords scope for aCCU1'ate research work. GENERAL. , Before proceeding with an account of the experiments it will be as well to briefly describe the phenomena of jets issuing from sharp-edged orifices, and to outline those portions of the tbeory generally given in text-books, which al'e necessal'y for a proper understanding (}f the experimen ts. , When, under the action of gravity, water issues from a sharp-edged circular orifice placed in a vertical plane, the jet so formed springs clear of the orificf;l, and, owing to the inertia of the particles approaching the orifice from points remote f!'Om the axis, converges for a distance of about half the diameter from the plane of the orifice, and then continues with a practica.lly uniform circular section, the axis of which follows an approximately parabolic path until the jet is checked or till the friction of the air breaks it up into spray. 48 The jet at its point of greatest .contraction is known as the vt1la con/racla or contracted vein, and the mtio of the area of the jet at this point to the area of the orifice is known as the co-efficient of con-traction-generally written Cc- the value of which is generally taken as '62 though it varies slightly with the size of the orifice a~ well as with the head. By the well-known theorem of Torricelli, the velocity of the stream-lines on the free surface of a jet, discharging into the ail' under a head h, is given by the formula V2 = 2gh It is found however tha-t, from various causes which will be referred to later, the actual velocity is somewha't less than that given by Torricelli's theorem, and may be expressed by V = Cv v' 2gh where Cv is a co-efficient, termed the co-efficient of velocity, and has an average value of about ,98. This value of the velocity cannot however be applied at once to calculate the discharge from the orifice by multiplying it by the area, for two reasons, In the first place the stream lines at the orifice are not at every point perpendicular to the plane of tIle orifice, bllt become' more oblique as one passes from the centre to the circumference, the central stream line being perpendicular to, and the outside stream lines in the same plane as the orifice. In the second place thi s con-vergence must make the pressure in the interior of the jet at the orifice greater than at the circumference where it is equal to the atmospheric pressure, and, by Bernouilli's theorelll, the velocity at the centre must therefore be less than that given by the above formula, as indeed has been proved experimentally. The rate of di scharge m ust therefore be less than that calculated from the formula q = Av = ACv v' 2gh where A = the area of the orifice and q is the quantity discharged in unit tim~. At and beyond the "Vl1la con/racia,' 'however, the stream lines are practically parallel; and it is generally assumed that there is little or no variation of pressure and consequently of velocity across the con-tracted section of the jet, and that therefore the rate of discharge may be calculated from the formula q = Cc Av = Ce A Cv v' 2gh where Cc A is the area of the jet at the vena coft/racla, Cc being the co-efficient of contraction. This is generally written q = Ctl A v' 2gh Cd being termed the co-efficient of discharge, and is equal to the pro-\ duct of the co-efficients of contraction and velocity. If Q be the total quantity discharged in a time T we have from the above Q = q T = Cd AT v'2gh It is evident that the co-efficient of discharge as thus expressed is easily determined by experiment by measuring the total quantity of water discharged in a given time from an orifice of known a.T,eft under a measured constant head. 49 The two other co-efficients can also for a ci rcular orifice be determilled .fairly simply though with less accuracy; Co by measurillg the diameter of the j et at the vma {on/mc/a, and Cv by deterlllini.ng the parabolic path of the jet; ill the case ('f jets frolll nOll-circular orifices however these two determinations are rendered more difficult by the fact that the shape of the j et!:!, at and heyonrl the vma (on/facia is not similar to that of the 9rifice from which they issue. Tlie water on issuing from a square orifice converges until at a short distance from the ol·ifice it reaches a point corre8ponding to the vma con/mcla of the circular jet. At this point the section of the jet is approximately octag·onal. The water issuing frolll the corners has a gl·eater convergent tendency tlJan that from points of the square nearer the centre, and consequently this water, in seeking the centre of the jet, squeezes out the intermediate particles of water. 'file water thus squeezed out takes the form of thin sheets, hereinafter terili erl rays, which lie iu planes at right angles to and bisectiug the fOIlt" sides of the square. 'fhe water thns displaced gradually recedes from the cell tre of the jet against the retaining action of the surface-tension, wi.,i cit preserves a connection between these sheets and the central portion of the jet. The surface-tension gradually overcomes the out-ward tendency of the sheets o~ displ!lced water, and then begins to bring· them back towards the centre of the jet. After a time the rays subside into the central stream and, as under a constant head the stream always travels a constant distance longitudinally dUl'ing the same time, the meeting of the rays takes place at a fixed point, amI the distance from the contracted section to that point, called a wave-leugth, is constant. But this is not all. 'flte effect of the meeting together of the rnys at th e end of the wave-length is somewhat similar to that of the (JO II-vergence of the water frolll the corners of the orifice, and the result is to sqneeze out a second set of rays in planes bisecti~ the angles between th05e of the first set. The second set appea s before the complete disappearance of the first set, so that at the en of the wave-length, 01' at what may be termed the node, there are twice as many rays as there are sides of the orifice. The secondary rays are subject to the same action of surfac.e-tension as the primary, and they ('on-sequently spread out and then converge again in the same manner as the first. Similarly a third and fourth or indefinite number of sets of rays may be formed. Practically the effect of atmospheric friction and other causes 'is to break up the stream into spray sooner or later; the distance over which the stream retains its perfect form depends 011 the head; and on the size of the orifice which is being used. This is the effect noticeable in the case of a jet issuing from a square orifice. When the orifice is triangular similar phenomena may be observed, but the number of rays is three, cori'esponding to the number of sides in the triangle. Generally the number of rays corres-ponds to the number of sides in a polygonal orifice; the extreme case being that of the circle, or "polygon of infinite number of sides when the 'Yaves merge into a single circular jet. 'fhe rays in this case are also Il)tletinitely reduced in section, and practically no rays are fOl·med. DESCRIPTION OF ApPARATUS. The foregoing phenomena. are displayed in a very perfect form in the hydraulic laboratory at McGill University, where the aut hor's experiments Wel'e made, and the following is a description of the appa ratus there used'. The experimental tank in the side of which the orifices are placed is of cast-iron, twenty-eight feet in height and fi,ve feet squal'e inside, the interior surfaces being perfectly flush. In ordet' to minimise t he disturbance creai;ed by the inflow, the water is admitted into a three inch chamber extending across the bottom of the tank, and having pel'-forations on the lower surface through which the water flows into the tank near the bottom. Twelve inches above the bottom is a baffle plate perforated with three-eighths inch holes, and six inches higher there is a second baffle plate, also perforated with three-eighths inch holes, the pitch of which was determined by the projections on a horizontal plane of equal distances on a sphel"e of five feet radius, with its centre at the centre of the orifice which is seven feet above the bot tom of the tank. There are two main inlet pipes, one three inches, and one one and a half inches in diametel', connected to the City water supply, and there are also two main outlets cOllnected to the bottom of the tank. The tank is provided with a gauge glass one and a half int·hes in diameter, placed neal" one corner of the tan k by the wall of the laboratory. On one side of the gauge there is a brass &cale graduated from zero point in the same horizonta l plane as the centre of the ol"ifices which is always a fixed point. Bebind the gauge-g.l ass is a mirror, while a carrier with a horizontal wire passing in fl"ont of the gauge is arranged to slide up and down the brass scale which is mal"ked in feet. Any required head can therefore be ob ta ined by bringing the necessary scale graduation, the surface of the water in t he gauge, the wire and it s reflection in the milTor into the same plane. On the wall by the tank is a ladder extending to the level of the top of the tank, and beside this is a vertical spindle pl'Ovided with handles a t i ntel"vals j this spindle op.erl!-tes a three-wa) val ve ' thl'Ough which the water J;I1ay be admit ted to the .tank or allowed to escape. The observel' standing on t he ladder could thus readily adjust the watel'. to the desired level. To obvia te the necessity of constantly climbing the ladder' during an experiment, a second indicator' is provided conflisting of a ~oat with a water-proof silk fishing cord attached passing over a . large light pulley and vertically down wal·ds in front of t he tank. The cord is kept taut by a weight a t the bottom and carries a ft-iction-t ight pointer which can be adjusted, whatever the height of water in the tank, to point t o a line on a brass pla te fixed in a convenient position near the inlet valves, which could therefore be regulated as required j near at lland is a small quarter inch tube with a tap which is useful for fine adjustment. . The head available on t he supply pipe was 280 feet , and the range of head might have been therefore greatly ext,ended had the tank been arranged so that it could be closed in at the top, a nd built stl'ong 51 enough for the full pressure. This would necessitate the provision of a mercury gauge. At a later date in connection with some other experiments, arrangements were made for admitting steam to the tank, so that experiments ' could be made with ' hot water, and as thi is always desirable and as the brass scale, when placed as in this case a t the side of the tank, would be affected 'by the rise in temperature, the author would suggest the desirability of placing the gauge-glass and scale on a • wall at some distance from the tank, care being taken to so fix it that the, relative positions of the zero line and the centre of the orifice would not be affected by the rise in temperature. A suitable vernier arrangement with scales on two metals, with different temperature co-efficients, would enable one to determine the true value' of the head with great accuracy. Greater accuracy could be obtained by the llse of a hook gauge which could be arranged for use in small auxiliary vessels attached to the tank at suitable heights and each provided with a valve to admit the water, when rea'dings were being taken at that head. I The, most noteworthy feature of the tank is the orifice valve which is a gun-metal di-sc three-quarter inch in thickness, and twenty-four inches in diameter, fitted into a recess of the same dimensions in a cast-iron cover plate, with gun-metal bearing faces forming a water tight joint for the face of the disc. 'rhis cover plate or body is bolted to an opening in the front of the tank, and the inner faces of the cover plate or disc are tlush with the inner surface of the tank. In the disc and on opposite sides of the centre there are two screwed openings, the one three inches and the other seven inches in diameter. By rotating the disc each opening can be made concentric with a screwed seveu and a half inch opening in the body of the valve. 'J'Jl e disc is rotated .by means of a spindle through its centre, passing through a gland iu front of the valve body and operated by a lever on the outside. Gun-metal bushes, with the required orifices, are screwed into the disc openings, and when screwed home have their inner sur-faces flush with the valve surface, and therefore with the inside surface of the tank. By means of a simple device, these bushes can be readily removed and replaced by others without the loss. of more than a. pint of water. A cap with a central gland is screwed into the seven and a half inch opening of the valve body anti forms a practically water-tight cover. The valve is rotated so as to bring the bush opposIte the opening, and it is then unscrewed by means of a special key proj ecting through the cap gland.. The valve is now turned back until the opening is closed whep. the cap is unscrewed, the bush taken out and another put in its place. The cap is again screwed into position, the valve rotated until the openings in the disc and tank side are con-centric, when the bush is screwed home by the key. . Fig. 1 shows the arrangem~nt as viewed from inside the tank. In opening the orifice the level' is l'otated through a quarter-turn till it reaches a stop form~d by a set screw fixed in a projection on the -52 r - - - - ' - - - - -J_ - - - - - - - - ..., , , ~ I , , - .~ I ' L _______ ~ _ _ _______ j ... ~. - ,";' .• ,1 .,. ... .... , ...... , ,. ""'J'~"'.-" . Fig". I. face of the tank, and adjusted so as to bring the centre of the ori1ice into the same level as the zero line of the gauge scale. The orifice plates were of gun-metal, the ceptre of the orifice coinciding with the centre of the plate. _ In making experiments on the co-efficient of discharge for small orifices the jets are discharged through a bifurcated tube into a calibrated vessel; iIi the author's experiments however, the jets were discharged into a !tume at the end of which is a weir, beyond which is a flap-door with bevelled edges along the centre lines of wJ,ich runs a piece of indiarubber cord, so that when the door is closed and pressed home there is a perfectly water-tight joint. ' This door is opened and shut by a lever with a spring clasp actuating a system of links acting as a toggle joint, so that the door can be rapidly and firmly shut and locked in position, each movement of the lever being recorded on the chronograph. When this , door is open the water runs to waste, but on closing, the door the w-ater flows over it into one or more of a set of five 1,000 gallon tanks of cast-iron, p.xe9- firmly in concrete below the floor level. Each of these tanks is cOilllected to a h eader by a valve so that they can be used separately or in any combination. To each tank is connected a vertical four inch brass pipe forming a float chamber; the 53 float is attached to a brass rod with a pointer at the upper end moving over a brass scale and attached to a counter balancing weight by a fine cord passing round a frictionless pulley. The scale of each tank is marked in 100 gallons up to 1,000 gallons, and then by 10 gallons up to 1,080 gallons. As indicated above, the duration of each experiment was recorded . on a chronograph connected to a standard clock in the adjacent testing laboratory, a mark being made on the record every second by a fountain pen. The time of opening and closing the flap door was recorded by a glass stylus the point of which followed closely in the track of the fountain pen, and derived sufficient ink from the moist track to make its mark when the lever was moved. The mark in-dicating the movement of the door was on the opposite side of the line to that indicating the seconds, so that there could be no possible con-fusion between the two marks, even when occurring at the same time or place. A stop watch was also used as a check against large errors -aud also to indicate during the experiment the time which had elapsed since the start. EXPERIMENTAL WORK. The water in the flume had of course to be kept up to the crest of the weir; before an experiment, therefOl'e, the orifice was always o ened and the water run to waste for ten to fifteen minutes to allow the water to attain a !!teady l)ead above the crest of the weir, so that at the opening and closing of the flap-door the flow at every point would be steady. During this time the inlet valve was adjusted, the chronograph record placea in position, and the measuring tanks read. The temperature of water in the flume did not differ by more than a few degrees from that of the water ilSsuing from the orifice, or as finally measured in the tank, so that no appreciable el"1"or was introduced, especially as the quantity of water measured was so large a ll d the temperature very nearly that of the maximum density of water, the experiments being made during the winter months. During the course of each experiment the head was kept under constant observation, and the temperature of the water issuing from the orifice taken at frequent intel·vals. The temperature was also taken in the measuring tan'ks when the quant ity was read, but no cor-rection has been made for the difference as it was within the limits of errors in the other measurements. This introduces the subject of the probable accumcy of the determinations. Q.- 'fhe total quantity was never less than 1,000· gallons, and in some cases was 2,000 gallons, these latte[" being in the experiments on the two inch orifice at high heads. On the scale of ' the tank 0'87 illch corresponded to ten gallons i the position of the pointer was referred to the nearest tell galloll mark by means of a steel scale graduated in hundredths of an inch, so that the combillea error of the initia.l a.nd final readings would probably not exceed one in 5,000 01' ·0002 as far as the relative values of the co-efficient in these experi-ments are concerned. The absolute values would depend up~n the 

The flow of water through sharp-edged orifices

https://openjournals.library.sydney.edu.au/index.php/SUES/article/download/2097/2365

The flow of water through sharp-edged orifices

Abstract

Similar works

Full text

Available Versions

The University of Sydney: Sydney eScholarship Journals online