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48
[DISCHARGE FROM oR1F1cEs
HYDRAULICS

FIG. 45

is measured towards the weir, (b) in consequence of the crest con- l or, introducing a coefficient to allow for contraction, traction, (0) in consequence of the end contractions. It may be l Q=Cl5/ (2g)Hi, pointed out that while the diminution of the section of the jet due When a notch is used to gauge a stream of varying flow, the ratio to the surface fall and

B/H varies if the notch is rectangular, but is constant if the notch is to the crest contraetiltln triaéiigular. £I'his led Professor James Thomson to suspect that the “" **'*§ ";§ "t:=»*f is proportion to the coe cient o disl" length of the weir the charge c would

A * m ', ,', ., ,, , ' /71 end contractions have be mucli more lé B ii-S li&;lra*=, l ~ll'H, ~, ;f.'§ i, 'lf', ;{/, '/.1-, lf Age? nearly the same effect, constant with;§ , § 3f;§ f;:;r f lg .ll'1'jl-llf:{l1;~{ ll., l, 'llf!11ll, l"l' l, ff whether the we1r is wlde different values 1; 5" 1

", ' ' gf” or narrow. of H in a trian- "11' '.'.'.Z"' "" f

% '7, ,»'{ 4; », f § '*l J. B. Franc1s'sexper1- gular than in a H % @1')'fh~§ |ll ll, Q 1.11-if ments showed that a r e c t a n g u l a r 31; ' ., ; }ifi, l'H;4lV~, l' l'., l |'| perfect end contraction, notch, and this 0 I

.'f;.!' '{, § fi1|, f, ?ll, ll l== - Hg when the heads varied ' has been experi- .L

5 " ' i" l i § "l'liiIl from 3 to 24 in., and mentally shown """' "'""-M

, ,.€'§ a=¢
;;E11e.-sei; Q% U16 1@11§ th Of the W@if to be the Case- FIG 46,
  1. Fig was not less than three Hence a triantingesd

thj headé diinin- gular 1no%h is mire suit lab lie for accurate gaugirégs Fhan ajrecltingular WV, % X 15 6 10 6 €Q '1V€ notc or a s arp-e fe triangu ar notcn ro essor . omson / 4 M length of the weir by ' found t=o-617. It wilf be seen, as in § 41, that since § BH is the an 211110L111t aPp1'OX1- area of section of the stream through the notch, the formula is ¢ mTm t 'L Wmvwwlww mately equal to one- ' again of the form 1 T' 4 1 3.122 -“Bids l'i;£'.; Q="<fBH><k“<2@H>" i=' '/=T "*** l 5 ' Cl =f§ s ' h ' f he ea vel it in the n teh to the //§ N, . ' where k, is the ratio o t m n oc y 0 —* —~- - —— 53? W (Ig thi? nolgchéw Weir' ang 4 velocity atithe depth H. It may easily be shown that for all notches QW/ b hi 3 Ca n?eaST1re the discharge can be expressed in this form. M '“ fi 1” th? Wcuil W 'Fifi § 44. Weir with ct Broad Sloping Crest.-Suppose a weir formed |l ll the 'Va-LCV lfgeir yfstlh ' with a broad crest so sloped that the streams flowing over it have a ll, ll', w i SH t? “ht Oh t, 9, movement sensibly rectilinear and uniform (Fig. 47). Let the inner M f§ Jet Pssslnlidtbrfiug tile edge be so rounded as to prevent a crest contraction. Consider a Q ~ ;§ ;“)ii§ ;'“fO1 etv;)0';1(i filament aa', the point a being so far back from the weir that the /+f'.:"'» ki ' ' 1

3:5-: ':1»»'1;~E='é. l, & contractions. In a weir 3, ' ' ~ - . 0

ra T., — *as ain, /%4. 'j, ; <1 1

"' ' "" ' '/ ' /f " l' end contractions. ff gs?- Hence, generally, the

width of the jet is l-o- InH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if CH is the thickness of the stream over the Weir, measured at the contracted section, the section of the jet will be c(l-o-InH)H and (§ 41) the mean velocity will be § /(2gH). Consequently the discharge will be given by an equation of the form

Q=§ c(l-o-InH)H/2gH

=5-g,5a(l-o-1nH, H§ .

This is Francis's formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value O~622, the weir being sharp-edged.

§ 43. Triangular Notch (fig. 46).-Consider a lamina issuing between the depths h and h-l-dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is xl (2gh). Hence the discharge for this lamina is

bx/E1 dh.

But B/b=H/(H-h); b=B(H-h)/H.

Hence discharge of lamina

= B(H -h) / (2gh)dh/H;

and total discharge of notch H

=Q = B~/(2.11) o (H—h)hédh/H

=f>'5B/ (2g)H§ .

iiii fiii

/ / Q "“S&.¢;g=, ..b"""

Ww%%WW, %rAl/r%o;, , t;, nMW/ //r/M

Fig. 47.

velocity of approach is negligible. Let OO be the surface level in the reservoir, and let fl be at a height /i” below OO, and h' above a'. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at (L is Gh”; at a' it is Gz. If v be the velocity at of,

v2/2g=h'+h”-z=h-e;

Q =be /2g(h-e).

Theory does not furnish a value for e, but Q==0 for e=o and for e =h. Q has therefore a maximum for a value of e between o and h, obtained by equating dQ/de to zero. This gives e = § h, and, inserting this value,

Q =o-385 bh / zgh,

as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry Coefficients for the Discharge over VI/eirs, derived from the Evperirnents of T. E. Blackwell. VI/hen rnore than one experiment was made with the fame head, and the results were pretty uniform, the resulting coejlcients are marked with an (*). The eject of the converging wing-boards is very strongly marked.

li Headi in Sharp Edgei Planks 2 in. thick, square on Crest. Crests 3 ft. wide. inchcsd 4 it I

iiltiriiilgihl miiiig-bdiiiiis 3 ft, long, 3 ft. long, 3 ft. long, 6 ft. long, ro ft.long, ro ft. long Water in 3 it' long' mit- long' 3 ft- iong- 6 ft' long' 1°h"l"ng' making an angle level. fall 1 in 18. fall 1 in 12. level. level. fall I in x8 Reservoir. 0 6o°.

I '677 309 '457 '459 '435 ' '754 452 545 467 - ~ 331 467 2 °675 803 -5119* '561 5155* -675 482 546 533 - - 479* '495* 3 '630 642* -563* -597* 569* ~ - 441 537 539 '492* . . . . 4 '517 656 '549 '575 602* '656 419 431 455 '497* -~ ~ 515 5 -602 650* -588 -6or* 6o9* -671 479 ~516 .. . . 518 6, -593 — -sas* ~6<>8* 576* -~ 501* ~~ 531 ~s07 513 543 Z; -608: 57§ ;' .. ~488 ~513 -527 -497 .. - - 5 I ' 0 '590 54 '470 '491 - » ~ - 4 50

9 530 -600 -569* ~558* -476 -492* '498 -480* 486 . . I0 - '614* '539 -5s4* -~ »- -~ 4651 455 ..

12 .. .., . 'g25* -534* . . . -467 ..

I4 .., . -49 . g . .. . .. .. .., .,

The discharge per second varied from -461 to -665 cub. ft

in two experiments. The coefficient '435 is derived from the mean value