IMAGE EVALUATrON TEST TARGET (MT-3) // r/j '% // ^ 1.0 I.I 11.25 S l£ 11120 M LA. ill 1.6 i^- ^ <^ V] /y ^ym / J^ ^'*.:> «^ ? / Fliotographic Sciences Corporation 23 WEST MA>i STREET WEBSTER, N.Y. 14580 (716) 872-4503 pening gives a more accurate value of the pressure than, a serh-s of holes; it can be faced without any burr, and Its width can be adjusted with accuracy. Also, the diameter of the throat at thi- i.oint where the pressure is measured can be determined with the greatest possible accuracy, and, moreover, is open to Inspection. The parts can be rotated relatively to tiie others without des- troying tile joint, an advantage in setting up the apparatus. It should be pointed out that the measurement of the pressure at the down stream end of the cone is not essential for practical measurements; it was merely used for determining tlie total loss of head in the instrument. The down-stream pressure chamber can, therefore, be dlspensetl with in ordinary, use, / GAUGES. The gauges used for measuring the pressure of the water In. the pipes are of unusual pattern. Fig. 3 Is a section, in which A is an iron base plate about % inch thick, provided with three brass levelling screws, B. On a ring of india-rubber on the plate stands a cylindrical glass vessel, C, or reservoir, about 2V2 inches in diameter, vv^ith an aluminium cover, D. Four rods, not shown, screwed Into the base plate and passing through the cover, have screwed ends and nuts at the top, by which tlie reservoir is pressed down on the rubber ring. Near the centre of the aluminium top is a tap, E, and a nipple, into which is screwed a T piece, the upper part of which is provided with an air cock, wiiile the liorizontal brancli Is connected with the pressure chamber by means of a piece of rubber tubing. Beneath the Iron base is a pipe, F, connecting the reservoir with n vertical glass tube. G. of ;■, inch bore and Ti.j inches in length. Behind tills tube is a scale reading up to 32 inches, and movable relatively to the tube and reservoir. At the lower end of the scale a spindle, H, is attached to the l)ack, and passes through a glnnd into the reservoir. The bottom of Pressure Column Srde Elevation Scale finely graduahd Top Plan thiH spindle Is of alumiuiuui. with a poiut ou tlie same level us the zero of the scale. On the top of the spindle Is a luilleU head, I, by meaus of which the spindle and scale can be moved up or dowtt until the point reaches the surface of the mercury in the reservoir. The'reading of the scale is then evidently the height of the mercury column above the level of the mercury in the reservoir. There is a email steel auxiliary scale, with .01 inch divisions, which measures the displaceiiieut. fi-om a datum, of the zero of the main scale, when the latter is moved until the point touches the surface of the mer- cury. The zeros of the small auxiliary scales of the two gauges are set on the same level. It is easily seen that the displacement of the main scale, or, rather, the difference of the displacements of the main scales from the same level, is a correction in pressure of water to be applied to the difference of the readings of the mercury colunms. The read- ings (»f the mercury columns were made ui)on steel scales, gradua- ted to .01 inches, and tixed with their zeros on the same level as the surface of the mercury in tlie glass tube, when the pointer touciies tlie surface of the mercui-y in the reservoir before the water is admitted. In this way there is no error due to difference, in capil- larity due to small differences of diameter In the glass tubes of the two gauges. I<:acli gauge is read l)y means of a Marten's telescope, ^^'hi(•h is fixed up at alwut 6 feet from the gauge and adjusted to tlie level of tlie meniscus. With a good light and steady pressure the readings can be taken to .001 inch. For measuring the vacuum at the throat, the pressure chamber was connected to a long glass tube running up parallel to the tube of the pressure gauge, to which it was connected at the top, where it was provided with an air tap. One of the difficulties to be con- r(f of unUonn biisi', has beeu sUowu by 08- borue Key Holds to bo glvfu by the (oiiuula:— r, := 0.039 tSIl where F,. = critical velocity. / ( r) = (I + .WMi T + .U0U221 2'-) -' T rr temperature centigrade, D — diaiiieu' of the pipe in inches. lu ussuiulug Ik'riiouilli'a Theorom \v(? are luaUiug the assuiuy- tiou that the lluid is t'rii-tiouk's.s, or else tluit the luotiou is such that tile ioss due to tlui(i I'rletlou luay l)e disregardeii. In the ures- eiit ease tiie loss in I'rietiou is large, being given l>y the values l\ I^^ Table 1. and siiown for different disdiarges in the curve Klg. lU. This loss will be divideii uneciually between the two coues. The loss in tiie up-stream cone will l)e niiuli smaller than in the down-stream, partly because of its short ii'ugth, but principalij be- cause the How is in general stable, while in tiie down-stream cone the divergence of the walls causes eddy motion. NOTATION. Let Q = total quantity in galionp. q = discharge in cubic feet per second. Jii, ^o> A:! = ^^^ readings of the gausses. (Jj, fjg, (5., = the readings on auxiliary scales. 2 z„ z — hei<'hts of the zeros of the auxiliary scales above the axis of the Venturi T = total time of run in seconds r =: tetnperature of water. IIo pressure of atmosphere. 7 Pitpt))Pz = prfHHures at the ptensure ciminber.s in feet of watiT. a I, Uu, 02 = areas of the tliroatH. •i» "o* ^i -• velociiiefl at tlie lliroatH. C =r coenicient of Venluri. g =r 32.170 tor Montreal. r = (lennity of in<'rcury. If wt' ciiii iiHsiuiic tliiil Hcniouiin's Tln'omii liolds tiHU> for llie motion ill I ho Vcuturl liio tiicoiy is I'xtrem.'ly 8liu[th', and ii sluiiile expri-wsloii can be olttaliu'd for tlu; diHcIiargi,'. Vakiii),' the axis of tho i)li)«' as levi'l, and in the datum plane, we bavo:-- Now P^ PjL + 2g Pi f'o - ■ - ss: U) (4) + ■ ah^ + f!, +2, 12 ' 12 12 //") + 12 .*. »(,- • - "r' = 2g H Now V , da a 1 H aav I 1 "0 «, Oi - — a" ■' 2.(7 H q = a„ i\ = "(I "I , J~r- -"= y 2g U To allow for friction, etc., we may put a a C' -. ^T-_:-7 '^2gH LIMITS OF EKROR. Total disciiarge, Q.~ln the gauging tank one gallon corresponch^d to .087 Inch, and the position of tho pointer could be read to 005 Inch, or about .06 of a gallon. Q varied from 04 to 330 gallons so that the error of reading varied from ,„',, to -,5',,;. As Q was measured by differences, the greatest probable error would be twice the above. All tlie readings were taken indepen- dently by two observers. t I I 1 Time.-Tlw (•liioii..Kni|.li wun oi.t'mt.'d tn.ni n 8taii*lanl clock In tho liilK)iutor.v, tlH' M..(k hcliiK conipMivd with the .Mh;||! Observ- atory i|„ok nich >Uiy Thi. ,list!iiKt- on tln« r.'cord (•nrii-8,...ii(llnK to 1 ■ecoiul was 0.4 lucht'8. ami r.'aaiMjrs ooKid i... made to .(Kjr. lucbt-a, BO fhiit tlj»' Kr.-at.'st |H.8slblt' error In a roadinw wiia ,'„ seconds, or tlw combiiH'd .mtop .,r tbo roadbiKs at start and llnlsh was ,' ■e('(iur mjK->dne readlnj? to .0001 Inches; tlie djjinieter of tiie tiirojit w;is (t.;5T!>S Irches: area^,- .OOOTST sq. ft; the diameter of the up-stream tliroat wiis 1 .CLiT Inches and the dia- meter of the down-stream tliroat was 1.020 inches. Mejin area .014411 acjuare feet. I I MEASLKEMENT (JF I'KESSUKES. In all except runs 7r> to H."^ inclusive, mercury wrs used in the pressure and vacuum fiMiiKes. .-nid readings were taken to .001 in. Tlio accuracy with which pressures are obtained depends not only on the reading of tln' fe'auj,'e. but on the settings of the zeros and of the pointers. The ievelling of the gauges could be done to within .01 inches of wat«>r. or say, .001 inches of mercury. The error of setting the pointer would not probal»ly Iv." more than .002 inches, so that the combined error of any single reading is not likely to he more than .005 inches. As the pressure readings were gener- ally very constant thioughout a run, the probable error of the mean would not be as large as this, and the error of 7', - jr. is probably not more than .003 inches, since ten readings were taken for each run. The vacuum, however, generally varied a good deal, and an error of .01 inches in the reading of /•^ + ]\ Is very probable. The greatest value of P, - 1\ is aliout 2.1 inches of mercury, giving an error of, say. 1 in S.OOO. Tlu> least reading for mercury was about .72 inch, giving an error of 1 in 240. The greatest value of 1\ + P„ Avas about 50 inches, giving an error of, say. 1 in 5,000, nnd tlie least about 1.2 inehe-;. giving 1 In 120. lu tile iVv,' exiterimeuts iu wUicii water Was used iu the gaugtsi lusteutl of uiei'cury, llio readiugb were only lakeu to tlie ueurest ,100 oi! an in, ii, and the order of the error Is iibout the fcame na glveu above. METHOD OF EXPElUii-L'-NT. As a general rule three runs were made at each head, iu a few cases four. The water was always turned on andallowod to how to waste for half-an-hour or lougt-r before lonuneucin^j: a series of runs, as it was found that tile vacuum did not reath i steady condition un- til some tiui^ alter the water was turned on. ^toy watches were em- ployed as a rougn checli on the chronograph aud for determining time of readings. DISCUSSION Ol' Tin: (JI!SERVAT10>'S. The observations are given, o-i Table I, and the value of C has been deduced from tlie mean values of each set of experiments, on the assumption that Bernouilli's !aw liolds for tlie cone. It will be noticed that c is in general less than unity, and Is least for the highest values of H gradually increasing with the diminution of head until a head of about 8 feet is reached, when It passes through the value unity. With still lower heads the increase in (• is iiiuch more morlved, rising to a value of l.;]58 for a head of 0.972 f<^ot. I'tO ng. 6 l'«9 1 40 I'M ISO K* 1 2B , Id l'20 Hh, \ \ 5? los V 100 s Vj tB ~~^ K>_. _-- — -« =a=s to ■ 1 1 'K i , HEAD IN FEET 10 I The relation of c to H is plotted In Fig. (5, and the wide vari- ation in tlie valne of c is clearly aijparent. It is noticeable that Herschel* found much lower values of c, but none of bis experiments give such a high value as l.iiGS. For moderate heads the experiments bear out the usual assuitip- tion, for this form of meter, that the constant does not dlfftr much from unity. P'or low heads, however, this does not hold, ariW a cdinpMratlvoJy large error is introduced by assuming a coefficient of unity. The variation in the value of c appears to have an Intimate con- nection with the question of the stability of flow in the up-stream cone, and on plotting the discharges as ordl nates witli the heads aa abscissae, Fig. 7, Fig. 7 OA ^ ■^ -»- ^ ^ Ojg ,^ ^ I To ^ > z 03 j(^ X^' o UI it y y^ y y t K UI Q. . 02. 1 i«_. y / .^ ^ / r ^ ^ 0flr- Ui UI IL ■018 lA / 1 1 ^- :^ i A 1 .^ K o 5 / _iiL--^ i^'' 7 / ,^ ^ ^ 3_ ,,^ ^ 1 00> A ... ,^ 1 f 3i : 4 6 Q t ' a 1 A 1 f 1' 1 Lo 1 H I:: , 1 ) 1 > 2 3 H 2 EA 5 D 1 N FE ET 4^ D 4 t 5 « 3 5 S _ It was seen that the upper part of the curve was of a noinewhiait different character to the lower part. This becomes more evident when logarithmic oo-ordinatos are used, for If the law concerning Q and H be taken to bo represented by the equation log q — log A- -f n log F and the slope of the line gives the value of n. * loc cit. il Ihis has been done iu Fig. 7, in which the line A B has a slopt whoB^ tangent is approximately 0.341, while B O has an inclin- a.^n whose tangent is 0.478. It. therefore, appears that In this me tr for low velocities the discharge is proportional to a root of H higher than the square and less than the cube, while for higher velocities it varie.s nearly as the square root of the head This rt^sult affords a clue to the rise in the value of c for low heads, for since its value has been deduced from the formula q = k H^ k being some constant, while the law for the lower part of the scale is represented by Where n > 2 < 'A it is clear that the value of c, being a factor of k, will necessarily Increase. ^ It is instructive to compare these results with those obtained by Herschel on a meter for a pipe of one foot diameter and on one for a nine foot pipe. These results are tabulateil in his paper, and from his Table I the values of q and H have been plotted logarithmically. For various reasons, which are fully stated in the paper, a num- ber Of the results were of doubtful accuracy, and, in consequence, are not considered here. The most reliable results appear to be those numbered 37 to 60, and these were plotted logarltlimically, and are shown on Fig. 8. i-ogq Fig. 8 r— ^ yA r ^ r^' \ — ^ ^ 4^ —^ ^ X" ^ y^ / X -^ ^ ^. y^ In this case, the value of n for the higher velocities came out as .49, while the low velocities of experiments 58, 59 and 60 gave a value of n — 0.6. 1-2 The reason foe this discrepancy is not cledr, but It may be poin- ted out that this first series of expenmeuts was conducted under great disadvantages, in spite of the care taken to ensure accurate results. The second set of experiments was made upon a meter for a pipe » feet in diameter, all the linear dimensions being approximately 9 times greater than those of the first meter. The observations were massed into groups, which an inspection of Table II. shows to be justifiable, and the mean results plotted in Fig. 9. . to • Fig 9 ^ \ »0 ->^ ^^ ^ .^ ^ ^ ^^ ^ ^f^ -^ l^ 10 ::d .__„^ AH the points were found to lie on a straight line, uaving an inclination tan-' \ almost exactly, and, tlierefore, verifying the theo retical law in the ease of large meters. These results point to the conclusion that in large Venturi meters the discharge is very approximately proportional to the square root of the head throughout the whole range, while in small meters the discharge does not apparently follow this law for low heads, but does so approximately for high heads. It appears, therefore, that a small meter would require special oalebration for use when di^icharging small quantities of water, since it does not follow the square i-oot law, and this renders the meter unsuitable for use in the measurement of, say, a domestic supply, where it is important that small quantities should be accur- ately measured, as well as lasses due to leakage in pipes, defective taps and the like. Moreover, there is a considerable loss of head in such small meters, as can be seen from the Table, Fig. 10, and this is a further disadvantage. On the other hand, its gi-eat advantage for the measurement of large quantities of water is manifest, in fact, it is the only practic- able method of measuring the water passine through a nmin. and it has been shown repeatedly that the loss of head in this case Is small. Fig. 10 CUBIC FEET PER SECOND No. of Expt. IIKAnS IN FKET OF WATER. Total time seconds. Total quantity gallons. Cubic feet per sec. Coeffct. C Up stream gauge. Po Centre vacuum gauge. P. Down stream gauge. 18 19 20 10.120 16.122 16.128 14.142 14.138 14.141 32.898 32.886 32.910 4.918 4.918 4.924 1,205.3 1,205.0 1,205.3 313.95 314.02 314.76 .0418 i .0418 > .0419 S .9494 21 22 23 33.098 33.047 33.040 4.838 4.841 4.839 1,205.6 1,204.7 ' ■ 15.3 310.46 311.09 311.30 306.44 307.36 .307.60 .0413 .0414 .0415 .9578 .9567 .9564 2;5b 24 25 13.174 13.101 13.100 33.002 .32.992 32.987 4.70G 4.708 4.714 1,205.0 1,205.0 1,205.5 .0408 .0409 .0409 25a 20 27 12.175 12.170 12.174 11.224 11.224 11.224 32.540 32.487 32.490 4.568 4.563 4.565 1,205.2 1,205.4 l,-'05.3 :-^01.84 302.47 303.85 .0402 .0101 .0403 2cS 29 30 .30.776 30.781 30.785 4.272 4 375 4.281 1,205.2 1,205.4 1,205.7 292.44 293.36 292.90 .0389 .0391 .0390 .9566 .9569 31 32 33 10.314 I0.30li 10.. 302 28.266 28.239 28.228 3.s^96 3.921 3.920 1,205.7 1,205 2 1,206.0 280.12 281.49 281.28 .0373 .0375 .0374 L4 No. of Expt. Head in feet of vater. 34 36 36 H7 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 50 58 59 00 61 02 03 04 65 06 07 08 71 72 73 Up stream guajie. 9.362 9.300 9.300 8 440 8.424 8.420 >M.417 Centre vacuum image. 25.813 25.791 ■^5.707 P, Down stream guage. 3 586 3. .58 7 3.587 Total time seconds, 1,205.0 1,204.9 1,205.2 23.519 23.090 23.527 23.438 7.516 J 20.525 7.509 1 20..')27 7.514 ' 20..544 0.549 0,660 6.571 5.03? 5.037 5.030 5.0;'-6 4.709 4.709 4.709 3.792 3.792 3.791 2.873 2.870 2.808 1.9i3 1.943 1.944 1.003 1.001 0.998 20 956 20.959 20.9000 11.471 1 474 1.479 17.726 17.026 17.029 15.154 15 165 15,146 15 138 I 12.325 j 12 223 12.138 8.955 8.916 8.879 5.879 5.853 5.843 3.382 3.324 3.288 1.012 .727 .759 33.183 33.197 33.079 2 524 2.494 2 438 3.220 3.267 3.263 3.258 2.821 2.820 2.825 2.4(;2 2.459 2 456 2.093 2.093 2.0itl 2 091 1.709 1.707 1.701 1.257 1 257 1.2.54 0.845 0.843 0.844 .506 .501 .498 .116 .127 .131 5.670 5.681 5.684 0,375 0.380 0.370 Total C»t)ic quantity feet gallons. ; per sec. 1,206.0 1,206.3 1,206.3 1,206.3 1,205.2 1,0.35.0 1,206.5 1,205.9 1,200.1 1,206.12 1,200.5 1,206.6 1,206.3 1,205.7 1,207.4 1,207.1 1 ,205 4 1.206.0 1,205.8 k,205.9 1,200.4 1,205.9 1,205.5 1,204 9 1,205.3 1.205.2 1,105.2 1,205.4 1 205.9 I 1,205.8 1,200 1 1,200.0 262.00 269.7'i 269.79 255.27 257.83 257,93 257.93 239.83 215,98 240.40 .0358 .0359 .0369 .0340 .0343 .0343 .0.343 .0319 .0319 .0319 227 40 225.46 225.72 .03fl3 .0300 .0300 209.89 208.85 209.66 209 54 .0278 .0278 .0279 0279 191.72 191.09 189.44 .0255 .0254 .0252 105.63 105.52 165.29 1,206.2 1 ,200 1 1.206.9 138.85 13S.59 138.62 111.70 111.32 110.97 74.46 74.23 74.00 335.00 335.90 336.15 .0220 .0220 .0220 .0185 .0184 .0185 99.89 99.89 99.32 .0149 .0148 .0149 .0099 .0099 .0098 .0447 .0447 .0447 .0133 ,0133 .0132 Coeflf C. 0.9619 0.9644 0.9587 0.9719 0.9704 0.9572 0.9800 0.9929 1.0296 1.1757 0.9537 1.0596 16 // HKAI> IN KEKT OF HATER. Py 1\ P. No. Up Centre Down Total Total Cubic Coefft. of stream vacuum stream time qimntitv feet Expt. Kauge. gauge. gauge. secoudfj. gallonss. per sec. C. 75 .941 .880 .552 1,206,9 75 12 .0010 7G .945 .782 .579 1,205 2 74.93 .0010 77 .944 .677 .570 1 ,208.2 72.63 .0096 1.2021 78 .944 .642 569 1,206.2 72.08 .0096 79 .944 ,034 .567 1,207.1 72.30 .0096 80 .775 .207 .459 1,207.2 63.39 .0084 81 .774 .195 .457 1,207.2 63.3.3 .0084 1.3583 82 .773 • .195 .455 1,206.0 63.28 .0084 83 1.485 2.314 .538 1,205.85 96.26 .0128 84 1.571 2.228 .626 1,205.80 96.72 .0129 1.0491 85 1.570 2.204 .664 1,206.55 96.72 .0129 In conclusion, the authors desire to express their thanks to Pro- fessor Bovey for the facilities afforded l)y liim for carrying out the work in the Macdonald Engineering Building. McGill University, 16