19. Professional paper, manuscripts, Flow over weirs with sharp crests and full contractions

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I FLf1 0 . . I

-·r g t r v ·l' 1 1 t .e ri d t b th m d _f ..Ll r -• r. _to_th _{i ..} _i "< in r th in r "' 6 :;, of _iri ,, t _ore!';t_"'e, n r b r 'V 1' r r n th nd n ... d •• f t ... ~ 0 -; _nc _ti _fo _h _t < d u un h j s -r I t .r of :t. i ~ lith

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y ons r ... l n T 11- r f 0 'l l

an

l;.c .. ual . - . e. c· 1 r t t c ti n l on i d t , ~ c i.n a"'

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un-for l v· rio 0 n ·on b ir

·o 0 J.h' y 1·· in . s _lJ. _c ₁ ... t .... ...,n+-·h l i .. r r· n Tl. ei r by y r· X nt .... 0 c·r _• -r ci of nt n 1' r ~ ur,.a. c y i on ) hi o·" wa 0 i lt t .n b ·n n I orv ~ vn th h 7 0 i . • l__-) r a t

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us th ~ r b ir e ) .. t ·)ur 0 Fr nc t

or

h :c .. , "11 y 6 to 2 1 t i no oul u f

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(9)

-3-Horton tat~

(l)

that F anci taa fo ul· he de f 8 r _ 0.5 t. to .0 ft. Fr ncia' ""'x .... r1.m nt re e y o tioualy e, but at

r

h,n

are r t eli,, ry of f Fr ci fo m 1 ha~

ill hold for

ly nd ·coneoi

n-r to the irr1ga or. ithout r gar for

Sub

the 11

weir d

hioh ~ nit, and it i n)t uncommon to eee

01 that fo:rmul~ for be de ab l.

1. f+. 1> r 1-ft i ₁ a d f

r

le th varyin ·rom 0. to 0 ft.

The .t .o

u ar

·e r h· n b tra zoi al ty e ·ith

ide a lop of 1

(a ) to 4J a a 0 ai th Ita

lf..'ll

eng:n

r, Ca. ~a.r

Ci,.; po i

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t t con of u 0 t t 0'"' 11 lim1nating th 0 rr ot on for e con r .. c a. f n~o.: y Fr-- cis c"nd u ec.. in hi f rmula1 pro ucine- a e

r:r

p rtion 1 to the l~ P"th

of the eir., and .. e-tng fre fro r

or

in ce of o e h~ lf p

r-cent fro y ain~l oau _• Th o and 0) of th t .a) -zoidal v eir t fu 1 con r'"'ct on n f. fall, he deri ed by a ~th a ical mo. fie tion of the Fra ci formula fo · he r ct._ gu ar

ir, · n o tain th~ val ee for the coeffioi n x onent by an e .inat on of Fr-noi e rl.m nt 1 ta, a ,. ar •trarily inor d Francis co ffici nt 1 e by l

ma.d.e a .e xperi ent ~ b t the for ~la pro o by hi w st .. te · to b ubj ot to th li itation i e by Frw o1 1 n th au -sequ n xten. ion of r n e of~ plloat·o of th for la n excursion into u xplor terri or_y. Furthermore, Cippoletti

(1) Horton,. Robt.E., U.'S.G.S. ter Supply Paper {f

aoo

1

p.~9. 4

(2) Cippol#tti,Caesar, C-nal Villore 1; o ulv er e D1spanae

(10)

-3 a

-da igne, th. ir or mini. ! disc r~e of 150 liters (5.3 ou.ft .)

er eoond1 nd maximum dioch rgc of 300 liters (10.6 011.ft .)

pQ eoond, hich, together · ith Francia' li it 1 restrict the use

of Ci·;; ole t i ' formula to ere at l ,ngtha of not les tl arr 3 fe ~t

(11)

-

t

-·ino h r io J.r 0 .c .1 ~ .e :.:or 1 ... 1

or . e. .ur ng ll 11 s of a r v r ir4!, 00

to re upon 'h en to ich l u

0 .... ocu.rately. ~ th 0 ... ._.ir i. n."'io 1 _.1

r ... "",c u DP' j ·rJ

r

n·"', ~hos r i -in Ct •. i at 1 on of t!'~. y ~ t h r f ioh ~ evn calib nJ 11 c l ... i e 1. , • or _t ee _{r.;; ··.son} C:l' t f.'\~,.;,. r ~ _o.n~ .u

:. or, _{Co. lin} _I

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_r'_do_., d i:n . 1.:;~13-1~~ 1 an""" +: r .ult

re J"i n vr ~

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( ) "cy. Tn l b r·~t y u.u ... s 'rc ~ .k 1 lrau . I a I construe v of et cy~ n n ~ c 0 r to l 6 /8 I ~,..- t lm.Vv ho . . tb e. ~ to b ... he g X • .~.;.--.n t _"bortor,. 0 h ... u. .:ly 1 ob or c t.,..r _• 0

-it'h aid 8..&..0 f to _I 6~ ..

t di .et .. r ~ i 0 _i cul h .. ~ I d • I il 18. _I

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_ch4. _1n ₁ _e_. I i ... ly \ '""

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-{ ~ )D, _{crib._,} _in _i _eri~ ev Vol. 70

11 .1-,P 6 u 10ct.

,..

I

(12)

-5-Th eirs are pl ced in th end of t. 1 box, bile on one si e of

the ir box a.re ·a . a, r b - t s n en thrv i e

is tl hoo g ~ stil box. The .~ter fl v~r the .eir int a

conor tv·- in or ni

x,

f e ·-,~. ft. io. an .. , ft. lon,

wbich i con: ,cte ·,11 .. ~n a.uxi lil;;i,ry or · .. ste re rvoi by o e chan.."'lel,

ann i t.h t b .... ca.li ~r-tea. t~ nk by ano h r

or

..

-tnnel. Th ~a.ter pas

:lntc ha! 1 ro b ~ 1~ i ~ 1 a r o .. e nt· . , ._._ ?~• 1·~1 ... er, io

are only by a st !el · l~t • aingl" di k on lever' a.r

openi gs. t elve- inch u~d a

fiv -i .. c rizo t~l centri ug

th tor g r 'he •.to

.J. r n in v ·on

o-tween th !lnor of .he c~ 1 ibrated r a.ux;.li(;\;ry re rvoire and. the oop" of e .... to e r, j rvoir i 1 · f ~ .:t.

So~ of h n t,. loy fo o· ring aoc cy 0

aewv tion a:ce,--The 1igh v laity .h n it leav·~ he

etora.gv erv ir he t th seri 0 ~f .,e bre u. he

eddy ourr n ~ anu r ti nd .· v otion to uoh n extent

tha by ,,hv tim th.. -t r <::1 . r ·he h,i box it ia in a quiet or

pon -11. c nd. i n.

.non i e o the ir b x is n.. llecl a.n over-)our

epi 1 ay hich re e oor 2 ft .. e hi ~h _ d 3 :feet long, hin:te

at ·.he b t he t i s 1

1 B ip-htly e to cf tt i l l y tn a. ;ron

of oil-c nvae att··oh~ to 1 : r box ani t e

.r

qo e of

the \..., or i uc · nno ... t no el the

ate to o r he ore t. -·ncb g tc 1' 1 Cl. the i

(13)

"

--the over-pour s ill -.: • B th by- A:\a:.e d.i cha.:rge into a. concrete box

hav.ng ~ tilv connec

:o

by · ect.na of ocre c ont ro1.. , and ..:~ .rJ. ... o a , lac i1 t, e en ~ of long rocl • r1 zm ing an a cc~ . .1 o

poe , i b e to keep t -- her .. d upon t e 1J e r t a 'Jo· 3ta.nt height

th:rougb-o·ut he u::rn:t. io11 o-r th :.xp..:ri ent, . .~. ic · t~.· t':r:or 40 t 40 t ·nutee,

The eltnrr t ion

whi.ol ·as :>Hilt ottetae o·. ha ~all of th1 eir b x and oonneoteo

ith ~he ~~1r box by 4 o e-tn~. piree G ·nchae lo ~. The etil~ box

is 10 fe t U1') .. t_ s.m fro the p an c•· the ·uei•.. It · e equip· ed . ith

a Boy·en type book gag an horen in the conor te,

yn

drop 1 -ht ~e "ts of c eful ea ing

thoue n ,h fa fo ,t.

A l ir .lr.-t.te er,. c

·or of steel· i h ~ asu ~ ee. Th ere t a_. i~e

n

el et:rio to t e one-ire y of brass ere .,:re ed to

trt1e an. le an1 str i'~ht l · 1ee n

A - y ~e ( t am orom te oal~~cr ~ere

calibr· ted to n.n :- lownbl .... <iiv-eraicj· ~ of o- ho of ar.. iAch

from a. f~ raigh line. The tria.ngu a.r notches ... re dre ... e.. o

tem-plate • A lla vy T-iron frame bout 3ft. h.P"h a.n 6ft long a

plac1.3A in t e concrete e l. . wall f ·. ir box, ·hi h as 6 11 chea

urf o ;;,d., bored fo 3/8 n bolt a.nd s ·1-rr~ n"'ed t.bat a. 1 steel anu br 'pln r1

the e1:t."S war att ohed, oould e ac j t d· a curate. y 1 an h joints

bet eel the '1\-f.rame and such plates a.ter tight

(14)

p t o , ith creat ace l t . r m

1 ~1, ro o d . 1 er r

f a fo

to

t • \ he i m ,r f ·o of t e

b : he·. · ~o f'luah .\';i.tb tl1e ere t.

In rr e tors er tho ·1

v

ion th 'eir re t t a

u th ti 0 t on -t o a

! a ~ oo , · .n a d

l e··.g-·ch '- tht1 1\.": s a hoo.. ...re · j -t t a the i nee fro hh

t n th 1 g ex c ly e 1 . t" _i -t .cc .... ·o the , t .... · oi t 1ng o ohe 1, n .ha j et · ng pl ·. to a zontal 0 th w 01.1gli t to t l6 ~3. e _h- ir t. ·a.t r 1 l 0 :lr bo· a f 0 f' ... of t. d t boo -g~ a. \ u l ... lJ el _G. te a.c y, th# ho k ,ag to 0 cr t h' indio t C~ a

ln be fl tua.t n~ c n iti _f flo o·

-tainu i 0" ing

an

t ats, ha b p i~e

for q c :J ::. '1<. nt th tan " 1 d te fl .... n 0 ained, as oc ed by mean .p a 0 ea 0 ·r l

r

....

0 era. teo. ) a cir d*s _• i 0 .,

.

u.te by .e· n of .el 0 )Q _r'"'o c· lly

insta.n 0 I y to v botn op

(15)

4~ flo- c. h ~ 1J.Y ili

h·· · .uen ~1e ~ t e handle

i ~o- ₁ h ch l ,.us .• o or in t ie on nd to The e.:~ o:.r fu 1 .. low o er iou- n

x.urimcnt a.t ,"·~ los of ··" u~ ti .. the

r

i · ·or 27' . f f ea.c i . ~ one l n

d. 'l : ~it, ns · ... s :re&e ~vo ... r

tic ,.1 cone e 1 • l • nk 27 t 61 _X _~71 1 \ o ta "' ro,..; i t o t.ut' o ';.., ... n l t o · 1 t ~.:: or a of ·a lktt y 14 c 1 c lr

e -loor l.re. rh~ e d. t . deep ₁ i l

l vation, n .citJ

,et Tho .na.ci y

so r.fnlly ~t~:trm·n3~ t;nd ··) c

l/1000 foov i e

vation~

b

fro. whic

n.

_ roun braos ro l inoh iarneter and t . ont;: 9

in e~"ch o librat :.C tank, being ~

a o t, G c by iro1 brC&.o et t t

?--

---)"

n · rv~le o about 1e r r

(16)

ho ... (;9-e; -. ie i l t.J ro . rov 1 l. Tit~ Cl" -9

-pin .hie anu- y

pl ... o ~dj o · "t

i the hole

o the b ~a

r. ~od a~ ·ancMOr d to he COncrota ad p·ov·d~

- CID:v to the "n- 0 ~ hOOK •

t h.v~lg 1• i·ch hoe in th~ bottom, rao ~tt~ch d ~ he

l~:tform .... n a, just .c.. to a atllline; bn.ein for th, 0<1 r -hen

+• · "• I he u t r 1 ..

ore

to h~ ne- uhou ~n t o

" t e.r h · e i n .. n ) a ~ c loae of each ex1)er: ent 1 t .. ,

0

t~t all !gin 1 v ir

rec"Jr L,

and .t:r.J pl_ cad .:n a

irb lu ring har crc'itG nd aide ,

ox 6 ft . deeD and 10 ft. ~i e, n t e

e·y 4~ ft. ~bo o the bott n 01 bha

OXJ t ~ t ... # '"Cir 0. '1.

1 1;), box Ft'Of i y to forty t~es

t o.... e ir 1 "'.u- "' · 3nt 1 var·a)le be ng th~ h .. d. Int .... r· a of or he d hvr._,1

• of 0.05 ft. u · dupll. ·1te ~ t

~. t .l

for · r.J:re · nt oi' ·, h , ' t-· froH u11l ic~ te te t

hie

o

itr"n ;)n~ .1'~-lf of o 0 ... ~i ra ult 0 erc~nt, o r p3r:tine th e ~ u t 1 uc -n re

• 0 c ura th ul .. id not a.s ur th

x t e indi idu .. ~l t t , ut it len~

the a.cour· oy o

o th~ a.et ~ot' o

ir-1

oy.

t ...

h orking oonoiti n » ~n ·ncrea e th

Co p· ~r t fa • t · ha t

e r

-run~

a l ' t of the xpar men 1 o niit ns n

o ... ~bility

hich in

-,he nic

(17)

I 0 0 l l 0 0 n l

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(18)

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(19)

BLE 1 PERCE G OF RROR I _{IS .. R}

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5.25 1.5 .11 1.6

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3 • 1.1 .l l . .l 1.5 • 5 lr.: • 1.2 ₇_. ₁_.₃ _.₁ _1.₃ 1.3 17. 1.3 _• l _• _• 1. _• _•

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1 g • l o+lo L+ l H. Th- l g ithm o p ot is a str ight

11

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,

h op n r _ ent the xpon nt I 71 ) ~

an 00 f 1o1 nt (n) .. cti l _• It i th r for i l

tt r ~0 _et _rn1n _h"

_{oorr ot}

_u _for _uoh _n _quatio

.

h n h ir t 0 in d in th Fo Collin~ _{hydraulic l} _ratory

r ic l y a f und t a.t t rr/q~F ~oto /crr

tr igh lin ut th nd tr p z ida

ou ch pr v 0 r r t y rath roo lex

ilu of ·rit o c r to hol to ig t

I l i 1 ho in t l

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a.n

co f io for C1p o t 1 zo1d. · i I h 1 " id lo 0 4 l . Th V lU· 0 h

pon nt

(n) nd th nd d or 1 h th l ol or r ot- n ul r () p 0 0 no Cippol=tti ir , l r 1r ur 8 b ins:ra fl •t r. Th.i t l 1 0 1 th OCU.l" C f th

im n al t _I ·or in lum h d.

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h 0 11 i io r pr nt ' th t r yp f ir re d1 imil r 1 r t diffe t m tho~ 1n I u t be r th many f oto h t 00 id t f ot po r m l l ir

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(21)

1r l Foot 1r .5 Foot

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. .... 3 .60 .17 1.G'

.

l.G~a 1.57 " · 06 ~.367 1.539 3.1!59 3.1 '"'.417 • • 1.502 3.326 ~.:...,a 6.1 0 1.423

.

5 .65 0

...

1.841 1.5

3

3.62" ₁_.₅₄₆ ₃_. _7o ₃_.₅₂₇ ₃_.₄₁ 5 . 1.505 3. 9

.

T l 6.930 1. 9G

.

3 0.70 • lo~.07 l .593 3.639 1.5:.~3 3. 3.95 1.5 9 .363 • ')7 7.768 1. cg -.~')8 0.75 ~.310 l . 02

.

4 ₁_._{5 9} ₃_.₅₀₀ ₄_.₃₈₁ 1.5lS 3."'59 8 .... 55 8 .. 662 1.501 _' .8 ::.5 2 l. l2

.

~s 3.508

...

1.516 3.3 0 9.532 9.561 1.504

~

.~

'SI

o

.

5 2.833 1.621 3. 9 ..,._ ... ,9 .... '"'12 1.520 3.Z53 10.449 l . 55 1.507

o

.

0 .l l l l . "'1 .676 •

a

1.52 3.356 11.3 2 11 . 10

1 .

u1o

o

.

5 "'.39~ 3.396 1. l • 76

.

'3 1.527 3.357 1"."387 12.388 1.513 .00 3.693 3.6 l .G50 3. 78 10.032 1.530 13.376 13.376 1.515 1.05 4.011

.

8 1.

_~!"'

1 .818 1.534 1 .378 14. 10 1.518 1.10 4. ~ 32 4.340 ~ 7 11. 07 1.r::.;a

15 .

67 15.425 1.5"1

-

1.15

•

l • l .r: 1 1"'.567 1 .559 1.20 . ·. _• 13.315 1.25 l .SO 1.35

f

t

·.

(22)

/4 Head ' '\. Feet 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 ~-foot notoh Dicchare;e Obc.rvcd

Currc

(n)

(c) Cubic feet cubic f et

per occor.d pnr aecond o.l52 0.287 0.453 0.653

o

.

aso

1.455 1.784 ::-,.14 9 0.284 0.4GZ 0.655 0.890 1.159 1 .455 1.786 2.155 1.530 3.492 1.565 3.?33 1.600 3.911 l .G36 4.066 1.6?1 4.174 1.?06 4.254 1.742 4-.284 1. 778 4 .303

I

l-foot notch Diechare:e Observed Curve (n) (a)

Cubic feet Cubic feet per nccond par oocond

0.300 0.554 0.866 1.216 1.626 z .079 t:..56!1 3.109 3.£93 0.300 1.498 3.327 0 • 5 54 l . 517 . • 4 24 0.8C5 1.~36 3.516 1.217 1.62;~ 2.072 3.111 3.696 4.,340 1.555 3.558 1.574 3.608 1.612 3.653 1.650 3.678

n ... t· rcr;ard:inc; dinohnrr;eo through Cipolletti notcl:eo u.nd exponents u.nd coe!'ficionto neoenen.ry

inntein6 the C1po1letti fornula.

1._-foot notch

riB cl a.1·e;e

Ob orvcd Curve (n) Cubic f ~t Gubio fe~t

p r oecond par n~cond

(o)

Z-foot notch

Dioahr rea

Observed curve (n)

Cubic feet Cubic !.et

p~r Aeoond per Aeoond

(c)

_r.i

charee ;;-root notch nte rvvd Curv~ (n)

Cubic f ot Cubic f.ut

(c)

4- foot notch

Diachn.rge

Cbserv0.d Cu~ve (n)

C~ ic !wet Cubic feet

(c)

cr P.l,.!conci. ,r uecond per secon' cr. eeoond

0.459 0.82'9 1.289 l . Ol 2.378 3,C.P, 3.718 4 .458 5.262 6 .138

?

.

om.

e

.

oz=.

0.465 0.8~ .... l .t-:::2 2.375 3.708 4.463 5.262

c

.

134 7.060 8.035 1.4£6 3. 2.109 1.49B 3.~53 1.513 3.413 l. 526 3.446 21539 3.169 L553 :L488 1.5U6 3.500 1.579 ~.508 1.592 3.502 1.605 3.~03 l.Gl8 3.498 l .G:'>i1 3.4R4 0.603 1.100 1.693 3.146 :$I 952 4.036 5.815 !Z I 792 7.043 9. !16 0.602 1.100 1.695 3.142 3.952 1.845 f> .Al4 6,844 7. 941 9.107 1.480 3.?.fi5 1.490 . ..~e307 1.501 3.35:. l • 511 ;; • ~~Hl 1.532 :..412 1.553 3.4;.A 0.909 1.633 2.536 3.523 4.6G3 5.890 7.173 8.574 1.5G3 3.42Z 10.081 1.!373 2.4l'f 11.85:: 1.584 3.4ll 13.347 l .G48 2 .53( 3.5r;a 4. 63!. b.8?0 7.190 8.!)74 10.080 11.655 13. ~$55 1.47~ 3.217 1.480 31263 1.48? 3.303 1.495 3.314 1.502 3.326 1.509 3.350 1.516 3.360 1.530 71.359 1.538 3.354 1.545 ~ .357 1.200 2.200 3 .3~2 4.?00 6.173 7.807 9.532 11.342 13,376 15.467 1.2C6 2. lfl 4.700 6.180 7 .7~5 9.532 11.390 13.376 151485 1.4 70 3.205 J.47G 3. 2~2 1.487 3.287 1.493 3.305 1.504 3.326 1 .F>10 3.331 1.515 3.337 1.5;11 3.329

(23)

/.S

C OF , ING ND D BO TO\!! aiD T lA .,.ON •

ti r ot n 0 r o te in eff ot o th d1 o

~~

n 90 J\ 0 0• t U h r nd C ppo tti ir t th 1 bo CLt rio b l

( l otto c ntr tio ) n th 1 iou t nc ut

f om th n o th cr n r u.d ' ( 11 nd co tra tio ) 1 ng h of l .O' 3.01_; _.. p i -f o.21 , _• n

l

.

O

•

r u d, a d 1 t n a o fro. 0.5' to 3.0' by inore l 00 p g u 1 of th discb r c~u by ) rc- tug - rror a to m·k th 0 0 • Th 1 ~ h th _i _c r o of t pth

=

.

a

•

n d o t n r llo in

c fro

flo

0 rr r 1 th c r '"' fo 3

-or 1 t n r l for r c ·n 1 ir • na 4 • e cur

-

-u 0 t i th ich ( ) r 0. I t 1

.

,

ir

or

th t 0 co t

o

.

s

•

i ch • 0. I

=

.

o

' •

T 0 io t hi h

,

( ) 1 (b) 1 1o 1 tity ich · y b lu , m1 u , r z o,

(24)

I(

.

o

a.o

2.0

a

.

2.0 1. 1.5 • • 1.0 l. 1.0 l. .5 LOC TY B .r.: ;.:,.5 • • 5 • . 5 ,.,.5 ¥ ~. 1.5

.

o

•

a

.

~ •

.

o

.5 1. r:: ·~ _.5 2. .5 • . .ABL J .7 1'1.66 q.05 • .01 .84 .3 8 .453 .5 8 .852 .2 2 .5 .2 .3 l .w 3.22 6. 3

er-e nt

Error • 1 . ~

.

' .65

(25)

• • _• .,..61 75

• .5 .1 9 3 .

a

• .?

.

.I. I'; 6. ;ool

s

7.01

• • • •

o

.

l . 1. .251 l. ,..~ .. , p, ,.. 1 cr 73 l . P: ,., 1 . • •

.

';)

..

.

.o .J 7 0 .304 1. .5 ll ,., _? ,"j • • . 3 1

a.

• ..1 4 . • '7. .518 4.20 • f &C:: •. 8 .5 1.

.

li? l . _•

o

.

0 .331 1.5 .oB6 5 .7-.; •. 9 .5 • l . 1 .400 .55 2 .88

.

f v •

• o

7 .501 . 9 8 .7 • -07 4 . • 8" () 7.92 l . _• .690 l . 1

.

o

.770 .826 ' . 5 ,.. lti.72 •

o

.

_• 3. _• .968 .07

o

.

0

.

87 2. Ll _. B"l

.

25 13 ~s

o

.

.5 .589 3.11 .v ,.. 6. .7'?.7 0

.

6

• o

_• .7';) 5.30 "'• .• "l8

o

.

1.037 9.

(26)

I

Tabl~~a!o ing v riation of Perc,ntage of Error in disc arrye

with R· t.:.o of Cro - ctiona.l rca of eir bo.x to Ar a. oD ir notch

for different nd n botto· contraction for 1.0 rectangular eir

with a 1 ac A 0.6 n~ ~lao 1.0 ft. Hea. ~ • 6 1 _Leng_h ₌_1._0' Side Bottom Out :Co n 0.5 .5 1.0 1.5

;;

.Q

Ratio A/a 3.65 5.32 6.9 .. 8.64 l.O .5 .48 1.0 7.96 1.5 10.47 t;.O 2,95 1.5 .5 7.38 l.O 10,63 .1.•5 13.94 2.0 .5 9.14 1.0 l3.BB 2. 5 .5 1' • 95 l.O 5.93 Percentage increase in Q .62 2. 5 .25

a.os

1.84 1.03 .74 .67 .95 .50 .40 .50 .35

.as

.18 H d =1.0' Length -=1. • Botto Do n .5 1.0 1.5

a.o

2.5 3.0 .5 1.0 1.5 2.0 2.5 3.0 .5 l.O 1.5

a.o

2.5 .5 1.0 1,5

a.o

.5 1.0 1.5 Ratio A/a 2.98 3.99 5.11 5.99 6.98 7.98 4.48 5.98 7.46 8.96 10.46 11.96 5.99 7.96 9.96 11.95 13.94 7.47 9.96 3.45 1 • 94 8.95 11.95 14,94 Percentage inoret sa in Q 8.25

e.oo

4.RO 4·.02 3.44 2.94 3.84 2.83

a.aa

1.82 1.58 1.40 2.20 1.58 1.3 1.13 1.05 1.45 1.12 .94 .82 1.12 .83 .72

(27)

/ 8

d t .- .0

· r o

(v+ b) . I

..

, ho · the rr"riation Qf percent 0

r

or

in di

om

r :>

:ti h th r-- "'"io of th oro - ~;;Otio 1 are of th ir box ( ) 0

t ·re of t ir to (a.) for ifferent n4 n bottom eon -t ot·on

'

...:or a r ot ngul r length of 1.0 f t

an h ad 0.6 ft n 1.0 ft. i l be een

0 n.- i g I) he po ition of the the air box hen the

bot to is in a ix d po it ion., as a great r effect on th dieoh( rge

t h nth aid

a

r

fix d and the botto i moved. Thi indioa t ot of n 00 tr otion t gr at r than the ffeot of

otto oontrao ion. End contr ct on a equal to 2 B nd bottom

cont ·~ction l to 3 H,

or

nd oont otio equal to 3 H a

bo 0 ual to H, a di o "Tg agr

ing

it t tO for

or

table for di o bigh he d clo r tr.An

p ro i · t ly l n ron , bee u e these di.en ion oau e, en

velocity of pproach o ou l/3 ft. per oond. These da a in•

ica e n l ooity of p ro h of l/3 f t. p r

al o abl for a l p roent error in i oharg • . distance equal to

a

H., t .. efor e~ms o b neoesa ry t f 1 ill th condition f

eo ul te eo t atio , proper, but ~n · ion 1 distance is neoa

# r o incr aee the or s - s otional area of th ir box anq .thu

r due t~e ·looity of appro oh.

By uperinpo ing th .. our ho 1 a the e f 04

of u pre

-ion f end nc bo to oontraotion upon t discharge ov r t ,a .

i:r a h e our f r the Ci poletti eir, ·

reo , ng 1

ar

(28)

11'

hould b t en fro bout th mid- oint of h si of the not

1 ste o fro h e d of the ir ore t. n r r to .- ·e

rvSUlt of the tv ty

r

r co

no t. e rror i i c ar r

box ·nor e t h , it is nti i 0

d 1 r

i thi th

to e th di oh r. f r the lig t e(;

a .. o. 00 l ..-l t 1 th ro 0 0 ble li t of rror. ( •J t t d,

EEl

In r r · h· t t i l d. i e o th i nd 1 · r i o· th. r

t t thy 'Y ro uce no ore eff ot u on th

c r o io

t 0 r ov

rvoir ( ~ir bo ),

i 0 r

rge

,

t

n

if

d t nee i .. fil t

r

a • H CO.tC fr 4

xp r nt t, t n e. oont _~ ion f 1 botto c ·

y provi te oo t r o io ~. ;

n c n r-oti 3 H· a .

~~0

H for _n _' H for bot o co

-traction A g t

r ct on ho

1 ot b confu d

r ·-oh, but the ordinary cone

r otio • inclu e botl ct on •

Th r i.o of oro eotio · r ~ o been _g·

~~

I y j j i t t ti n o r 1 t h M >tha ac

c n

-eff ot of loci of t t •co · 1 t c. -of . eir box to o (S) co tr ot.o n ro nt of

rror of di char vari "th the r tio of d t tt . co t:t-- o -t ion i 'nee for ach r·tio of A: 1 1 dic·tin no fix

of the r·tio 0 gi a o n t nt ero of rror,

and that tl v lu ho 1l b r ·~t~r.th 7 1 n c e,

ro

(29)

{1}

Fl~noia, Ja. e B. Lo ell Hyd ulio Experiments, 5th Edition

" g s 12 n~ 134 •

Hydraulic 1 P. 120.

{2} .. ith,

Ra.

ilton,

Jr.

( ~ Ci pol ti, C r Canal V1llor

Gippole~ti ooept d the r ault

i~ilan

less}

P. 23.

of Franoi

t

experiments for n nd bottom aontr otione. He also quo rule d duoed

{ld!m)

by Le bros fro .. th r ul of hisl\exn ri ent # tb t both

con-traction should be t leae 3.?0 times the d pth of the p

-a fr m the experi ant of Fr ncis, Cippoletti d duced the

follo ing; (a) hen th nd oontraotio i e ual to 2 H and tha botto contr otion equal 3 H, t a 'bot"o nd 11 have no long r a .. a.ppreci ble influ noe on th di charge of the eir.

Th1 o n ition mayo us n inor a e o. bout 0.15 p roent.

(b) itb n end contr otion of 1.5 H nd botto oontraot1on of

2.5

H,

t inor a e in c11eoha.rge ould b about .5 percent.

(o) .If the end oontr otion i l Hand

th

bottom oontractio 2 H,

'\h di eha. g ill b inor e bout l pero n _• He al 0 t _ 8

coount of the f that th v locit of pro ch ha.ll ot

exoeed a oert in 11 it.

(4

Smith, H~.: ilt n,

Jr.,

Hydraulic , P. 122.

(s

Tlle ooeffioient u ing th - expre ion of ratio , as

pro o ed by J. 1 oh 1 1845 and h b n elaborated uoon by

ny. Sa Forohh 1 er, Philipp Hy r ulik, Leipzig, 1914, Pa 312.

l6

C rpent

r,

L. G. Colora o Exp ri nt St ion, Bull tin 150, 29.

(30)

In or _-r o

thro urth r ligh u n tu 0 th .> c ott 0 0 ,

-tr~oti. n upon he g y~~

·

tri n r notoh

(*),

x-P ri e m th

~ ~ving

th J. oar tle

~eir bo l v l iv tn~ v rte • I e c· e t f he

ir box l ft.; th :t.me a in th

:p ete co tr· ti n· , ut i th tand ... 1 e f oor b

4.5 t. 1 :to r rt x.

~~

900

11 nota ~ · i th o to oontrB. t"on

d, 0 t o r P nt y th

rm

E.. eou i rly,

_ro

_~~

flo ov r th· o~, notch h in . 0 : 2.53 H • 6

'

:0 l i # Tho(p n'

or

th

oontr ti n • I i oba le that Ole art f th inc

o t in h n th floor l le -1

v rt u 0 incr r ed looity of p roaoh. ~i inc e ··

in i ch· g a ount 0 1.6 f r ad of 1.0 f an y

i ini~ he· th head ' or· ed. T rc e i i

re!)r y th for ul E

=

l l • . H • 1 - 10 _I hie 0

not ho elo a e.d qu l to .3 ft.

IL f 11AY1' '

(31)

::2-/ 0 .h a 0 f t :ft. te _• r 0 ( ) n P. ( )

(32)

t of a out 5 r t

/"A

_• t 0 no f 1-- e

or

t h.i ita u i u o .. t r d t · Q M ,.. - gul.v otoh irs it h l 1 l' t no no houl b· u e t gr t t n . b u _• _• "" t 0 t .tt?J not h

ra

:o f t i 0 r1 n t h ir

to

j -4:1e5

•

= (. + ~. ~ _S) (S) i v 1 0 h i e, l.t {H) "'d • I d t v <..\ l q e

~~

~

note·. -..~h01 1-- _,a. a _- .2.u • 87 00

r

0

r l 0 u a. 0 e.

~~

T r 1 r o- e 1' 0 th· 9 A to

'

a -• I l' in co and r n g t -r t . It 1 l 0 e of l h· 0 !J'. ith ~ h

.

0 0 r

.

o

.

,_ l r i h c cr .

_•

r 0 1 1

'

he h r •

....

no c t e p a. t 1· f

r

rt. 00 . i iol I j t

(33)

du~. 1 fo v 1 a, ..

r

ula. y ... ai )ver-,uncl in Ta l

:: .49 Ha. 8 1bioh giva c isoh~rge

l aely 'th tho ~ by the

(34)

TABL 5.

DI CHARGE ABLE FOR TRI NGU R NOTCHES

Co pu .. Fr th For l • Q = ( .oas t-2.462 s) H

(

g , ~ 5 - .01 s .75 5 )

- -

-

-H-

ad

l

-

/

- - -

Y.

Slo

~

- - - -

-- -

-

- -

--

-H Ft . & ln. ₂₈o 4, 30o

soo

soc

J >

.

ao

.21

.a

a

.23 .2 -"' .25 .26 0-3-

/8

.

.a?

o

-3-l/

_

.28 3-3/8 .29 -3-l/2 .30 .31 .32 .33 .35 .36 .37 .38 .3 .5 .5 .52 .53 .5 0-3-5/ 0-3-3/ --o-1 /1 .. 0-3-15/16 0- -1/lS

o

- -

,

/16.

c

-

-5/16 0- -7/16 0- - /16

o

-

-ll/16 4-13/16 0- -lo/16 0-5/l/16 0-5•3/16 0-5-1/4 0-5-3/ 0-5.-l/2 0-5-5/ 0-5-3/4 0-~-7/

o

-c~e-

;,e

o-.F-1/ 0- - /8

o

-

s

-

;a

0.012 0.01 0.016 0.018

o

.

oao

o

.

o

2 0.03· .026

o

.

oa

0.031 .03 9.037 0.040 042 _;OJ'(~ 0.046

o

.

o

9 0,053 0.056

o

.

0 0.064 0.06 .072 0,077

o

.

81

o

.

o

6 0.091 0.09£ 0.101 0.106 .11 0.11 . 0.123

o

.

la9

.13 0.142 0.013 O.Ol& 0.017 .019 0.02 0.023 0.025

o

.

o

8 0.030 O·OS3 0.036 0.039

o

.

o

a

0.045 0.049 0.052 0~056

o

.

oso

0.064

o

.

osa

0.073 0.077 ;082 0.08?

o

.

o

2 0.097

o.1o

·

0.108 O.ll Q. -19:/ ;J,o 0.126 0.132 0.1" 0.145 0.152 .• 027 0.031 0.034

o

.

38

o

.

o

3 0.047 0.052 0.057 0.062 0.068 0.074

o

.o

ao

0.087

o

.

e

0.101 0.108 0.116 0.12 0.132 0.141 0 .150 0.160 0 .17{) 0.1 0 0.1 0.201 0.2 2

o

.

a

4 0.23 0.2 8 0.261 0.27 0.387 0.301 0.315

o

.o

0.052 0.05 0.065 0.0'12 0. .!o"5' 0.115 0.125 0.136 0.1 7 0.159 0.171 0.18 0.197 0.211

o.

-o

.a

4o

0.356 0.272 -0.2 9 0.306 0.324 .343 0.362 0.382 0.403

o.

4a-0.445

o

.

68 0.491 0.515 0.5~9

(35)

.

so

.61 .62 .63 .64 .65 0~7-13/16 • 6 o-?· 15/16 .67

o

-

- l/1 . .a -e~3/l

• e

-

s

..

l/4 .?0 w3/8 .71 .,;. /· .?· 0- -5/S .?3 0-8-3/ . .?

-

8-?/

.

so

.81 • 82 .83

.

a

.9 .9 .92 •. 3 • 4

.

~. • 96 • ~ 7 .99 1.00 1.01

1. oa

1.03 1.04 0-9-5/

-

9 -

3/

o-9-13/16 .. •15/l . 0-10-1/16. -l l /16 -10- 5/l ' 0-ll-1/16 - l --3/ 6 - ll-1/ ~ 0..11--3/ 0- ll-l/ . -ll-5/

o

-

11-

I

0-1 -7/8

-1-

l

1-Q-~/ 1...0-3/ l-0-l/2 C.l84 0.191 0.1 0.207 .215 0.268 .277 0.287 0.29? .307 .317 0.3 7 0.338 0.349 0.360 0..4 0

o

.

3 0.456

o

.

69 0.4 2

o

.

5 0.50 .523 .53 0.55

o

.

41 0.656 0.672 0.688 0.705 0.23 0.248 • 7

o

•

0.276

o

.

a

0.296 0.3 0.317 .328 0.339 0.350 0.361 .3?3 0.385

o

.

0 0.473

o

.

7 0.501 0.515 0.52 0.544 0.55

o

.

73 . • 9

o

.

8 0.702 0.719 0.?36 0-75

so

0 90° • 3'0 - - - n.o~

-0.345

o

.s

0 0.360 0.617 .376 0.64 .. 392

o

.

?2

o

.

09

o

.

26

o

.

4

o

.

4 a

o

.

8 • 99 0.618 0.537 0.557 0.5?8 0.599 0.620 0.642 .6 .687 0.7 0 .734 0.758 0.782 0.807 0.833 0.859 0.885

o

.

12

o

.

40 .968 0.9 6 1. 2 1.05 1.08 1.11 1.15 1.18 1.21 1.2 1.2? l .:il 1.3 1.38 l. l l . 5 1. 8 1.53 1.5 1.5 0.7 0 0.730

o

.

'7·60 0.790

o

.

e..,a

• . 54

o

.

87 0.921 • 55 0.9 l 1.03 1.06 .. . 10 l .l l~l 1.22

1 .

as

1.30 . 1.34 1.39 .43 1.4 1.5' 1.57 1.61 1.66 1.?1 1.7 1.8 l . 6 a.:t9 2.2 2.3 2.37 • 2. 2.55 2.61 2.68 2.74

(36)

E d H ad in _{o ,}Slop _~eo _o 90o Ft,

&

In.

as

4 ~

eo

'I.-

5 - -

1 -

o

=

s

?i----

o

.

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(37)

i charge ii1 i

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(43)

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1. ~9 l. 40 l.41 1.~12 l.43 1.44 1.45 1. 4;. 1.47 1.48 1.49 1.50 -l •1'

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l'>-15/1· 14.-l/16 .-3/16

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14-3/r 4·1/2 1 -5/8 4-3/4

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16-13/ 6 lr)-l.:/16 J,.'7-l/l.<i 11-3/16 11-l/ .4 1'7•3/ lt-1/2

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(44)

T t i-oh ir l ~~~~~~~~~~~~~~~=u~~~~~~~~l=a~s~; Tbe in io t for t ol n or r ota.gul r

ho.n i gr _.io a~ t b lar o 1 Fi •

t . i b n f t o ... he e curve n th tabl I t -t c.. p _. r 1 m .. t agr e f i r 1 y · G 11 , it h ~ h r ci for u i hi e o H =

!! •

3 Furt ~ or .. ~ _t _{r .. oi} t th fo .

.J;:~

1 i r io f l o 3 o h d t 1 n p · rc nt '.,he th s no e oe Ho these t~ u. no i .~.d o e n·, o 1 if t th re proper for r a.t r po · b i u Q. ~0 00

cour oy for h higher . a

.. <" !o i .. ~~h o ur~ ; -an,. h l i ita.tio of (H) to (L} / 3, ie a.pp r ntly n o ar o

th

1

i

ne ·

0

to io it " pli , but · ot u o a.ny

ir. I ·Oth r ord , the 4i it v· u

ad on t e fo nJ tl£;~., but h hort-oomin. i 1 u th

the ot inheren i tht: eir. Th u o of the

p her in, o only pro i a a g eat

r c but 1 o ex nd th · li i

li its o tle rat·o o h

of u

o or

of air • The

leng h to io the e

for.~-.u · ··pply, i ot ~no n, b1t hey hol for t e dat for the

1

t

ot ir f r of l • J

of a to e 1 n, h tri

r a.ll eir oo_i

n fo h d 0 oot

0. 5 f t • c r t 1

-

1

gt __ , bu..: t h

i ffar nt formula. For all o th

hich tl r te t r ti

ir to bio t~e g n r 1 ·or l

it t 1 o t1on of th

r 0 i

for t ir oll

(45)

3 S

Uio~r por ion of the di charge

ourv

are u1t c n isten~.

Th ne\t formul i ""'jore comnlioat a than th old on 1 but ein e

t bl~ are g

ner·

lly consulted ford ter ining the flo o er

ei1 .. s# e·reci lly wh·'n delivering ~a r to the irrigat.or, tn

.... r o . · oa.l o i$ .... dvu.nt;..g of .Lh~ nt: !or tula. i largely ov ·roo .le.

·hen nd i c l'g d to u e forrula to~ co puti·l _ t i ·

-c · rg 1~ tl · fi ld, an pproxi ion

*

lly u!fioi t , in

nich cr th ol f )rtdUl a i uf ioiently accurate f r ·.ny

-:;;c ical hea o ~r th.., 1 ir. Although w ir t ble bcul"

be b .. i~d upon -.;h moe, a.ocura,te for ula. a a.ilabl , the oofl

-·uted d~ oha ge o le no e xpres ed gro ter degree f

(46)

*

R 1 r y

-

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-6

-fo

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-l . J B _•

.Jo

L R CT

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H

.

99.5 8 • p .6 98.

(47)

C

_ _ _ _ _ _ _ _ .... · Tr p zo1 1 _ir- _v _i _lop _of

1 to 4, _ d i y the It 11 ngin r Ca ar C

are v ry ext si ly u i th irrigated t , an

oletti,

re ther

-for _ p i nte u n. T 0 1 crest 1 ngth r 0.50082 f ., 1.0050 ft., 1. an th e fiu re ft. , • 002 ft., 3.001

tt

.

,

4.0058 ft., no in l 1 ngt .... inc th ei iff

d t oug out th o lc,l tion , the u ed mer ly or ref re c urp-oe

in h r of

c

p ol t i nd re t

-t l t 1 re r .... s n a

sid lop

,

or

ro

ly

gul r . wir otoh of ual 0

tr i- .~l r notoh b ing to

a.

aao

4 t .. !lgle, th di obar e re oter _ind x~er lly for

uo

'l

no ch. It fo n 1 o r~ t t t 1 char

er

this

n t not x otly eq 1 th r no b t i oh r

fo · C • pol tti n r ot n

r.

•

'rh diff r~nc _b

t i o r~ tor Cip ol tti ir - -nor , . it

th or ll 1 o w ir ; t r r ula i

o

r

be p to diff r nc cop r o

.

- ·n i r Cip h h Fro p

r

n

1

or

in ut for c e t le . ~ r in t

o

.

t

or

1 rg i • I 1 bar d

or

h 1 ngth inor~ .. : J Q D 1 i le

tor

t .; . t eli o r o r t l to lo iff r no b t ngula.r w ir , it n i nth isobar. s for i p ro nt ge of x

,

nd qul z r pro i 1 2. 't • th lo

or

oa e ~lote of the eri nt 1 d ta th dif

(48)

f•-r no 1 1 cb g ov r r ot 1 r n Ci 1 t i air for EtC O.l t . h d, .d ch 1 n tb of ir r it r oh h th

a u

of thea d f f r n tor th r th r n o~t l gari 1 _-· in t t fr rif th 00 h lu 0 di f r n r-- d to j r en d by th· tio Oip. _• _{- Rect}_. = .60 B

•

Tl1 r fo ,~ d i hi t r to e g n 1 orm la r h di c r r re t ngul

r

ir

,

wil i t r f t

e o e Cippol tti e1r

'

hio 1

-= 3. 47 1 H 1.48 _ 0-.568 L l .S H 1•9

+

0.609 H 0

1+ 2 L

1 .

8

'l

ov r Cj··) _ol _ti rs co

-put:.d

ro

the abov fornul

'

a in

T

1

ta>

agr e ith

th xperi .ent 1 ithin _{P ...} t l _• 0 t . T

error or j cr _anoi re _·₀ _it_i i~ _c _c nr- _n _'"' _ti"

i ot r· •

T Ci~Iol tti eir h vi g no i o e t l 0.5 t •

i - 110 o r _fo O'u i th 1

~-jr , o iuly forth re on- ot 0 'o _et_ for th .5 ft.

r ctan ul r >! ir. I ul b disc our e n f vor of

th' 90° tri no ch. T ne ol o ng fflr ul re .r nt t

1 over ft.

c

ol tti ei _I an 1 fc t c io r 0 only

-=-1.593 H 1.526

_{l

_t

₁

₎

₊

_•

87 H 2.53

(49)

T 1 st .... r . of th abo fo la r res n't the

differ n e in 1 c r bet e n th Ci oletti and r ot ngul r

.. _i_rs _ith₀_.₅ _f

_or

_t _le _tb_• •

Cippoletti ir do not iv ~ dieoha.rg proportional

to th i ·ngth, hown by th di ous ion on Pa~a ₀ Sa • _-

(50)

Weirs nts r -:e \- ith eira

having oreat l~:gths of 2.0 ft. d aide op~a of ~ to 3, ~' 1 tv

flnr kno•;l._,i .... e o..: the eff-..ct f side alopaa on propoztiona i·~y of

dj.soharga tc 01"(~·-· -4o length, lf.-.lch lS discussed on Page 0

o

_.

61nc3

air 1 but the diaobarg ov r thom i abo n

gr~~· hioally .in F·g. 19., ··bich also inolu os th · c! ree over

tha l to alo e · nd r otangul -~ · eir for co 4)arat: vo Jur.. e&.

b h old nd ne · for ulc\,S for Co.-•- ol tt:

~1ai:r-·hjcb it ~ 11 be s er that the

rror

o net exceed

a%

ithin the 11

it

of tbc rut io o:t 1 to 3 of betird t.o length.

ne·· forutu.la. ano li.itz:.:·c;ions of the ol fo:rJula. for reo a1gular ,.airs

7

(51)

T BLE 10. CO. Riso· OF OLD AID NE _IR _!~OR.

_JL.o.•

CJ

P

L TTI

ErRS.

H- - - ·- - -- - - -- -

-l Foot ir 1.5 Foot ir 2 Foot c ir

in e 0 01 N Old F t _ _ _ _ _ _ _ _ _ c __

j

_____

__

_____

%

___________

y_

.20 .302 .33

.so

.67 .85

1.oo

1.25 1.33 1.50 • R 1 tio 1 o 1 ho n by

:o

for ul o b .301 99.7 • 50 .63

s

.1 .c 4 _• 1.19 97.5 _1.7 _1.7 95.9 2.81 a.77 9

.

s

.07 3.97 9 _• 5.25 5.05 7.49 7. for ol nd ~ or ~ula 1 t in v·lu o e 100.4 .599 _• 02 100.5 l 0.3 1.27 1.38 100.8 9 .• 4 ₂_.37 _Z_.₃ _100.4 9 .6 3.70 ~.69 99.7 97.5 5.33 99.1 96.2 6.86 98.1 9 .3 .72 96.8 3.0 oo+ · ir Je (')1-n _1<-t. ~ _• 100.4 101.1 1 .1 5.41 1 o.~ 7.97 1 0.5 lO.Oi1 100. 9 .3 4.0 Foo ir li . 0 d

..

~~~-~~-~-~-~--~-~-~ 1.198 1.205 100.6 2.52 ~.55 lOl.E 4.69 4.75 101.5 7.2S 7.38 101. 10.42 1 _• 101.2 ~3.33 13.47 1.1 18.72 1 • 2 100.5 20. 8 20.65 100.3 3 .'15 .74 100.0

(52)

CIRCUL R EIRS

.f J. .. arant y no e h 7 ev.r ;3 11 ! t. d. u ... ni

-cul r ur 1 c ... roul't .... i!' i .... ... ver ic~l ., 0 _it_iort_with} Gtl(l "

th i m ... r or·th O~.J r1in _• I r er to thro li;tt

upon t'- ·9 _..rot bl di-,oh rg throug ir~ ,(>

..

thi~ ha Je, r: n.: "'"04

r t ...

-1hen oting ir a IlOt ,. an

or

fi(,

,

exp ri.dl1"lt ... . e.r:; n

on Q . :tp cr ote ciro 1 "' ir of (). 95 f ,. .(')025 f t . i· ....

-.eter. I an .. emi-c rou r . ir of 1.5011 ft . 1-. goo .r..t. i

-t r _• The iaoh· rge d r nl tt 1. Fia-. I hie .- 1 0

th exnvri n"' 1 d.,.t·-. vitr1 t. ;:; 1.

-

on_-.t-or ula.

D of 1,

_•

_ir_{_Fo}_r_mule_; _Owinz _tc. _th ₀ .

-ely i :t t "''. X" e 0 c tR. .. it - 9 no 0 'lee·· r

hie 0 r t th~ ""* ;_ oh- e for t~o '.1 1

o.rif·ce \i..nc~ ei action₁ anc. · eitii 1 r tyr>e o formul~ · l: d'"'

uc-oe - (D) . H ~ r, i . the ~en ral I

eme r a onabl to aurpo that (n) ill a ::ronc.l

• 5

±

it 1:. inc r ' i .g Vf .. l u _o _{H/D ··} t e our i a.··

r.

ac

t n :;;: 0. 5 ± 1 and. a n

=-is '\Jgative, o i 11 h > ,:1 • O:f r n) ..

t itl Q t

ro·

the greater heads, it ould ~e po to xpr

r h t it oul' not i fo.L ny but

·ould • . c h.; 1 .. b xpr a i t rms o (D) an~ ~ ratio o~

H/D in """oad 01 in t rmn of (D) nd {H).

(53)

t oh i of _ ir n( .jlott .- arrE,inat the r ti.on H/D. Fro tl11

~- .~..o t _ for ul for ( ) c a obtain

n=l.93- 0.3?7 ( ~+ O.l) 2.57

hio h 11 u . .J o pp oxim-t ly B qual to D. Va.lu

ro. th oir"i w r ivi th

to ~~i ,e t v l e of (c). T e v~ lu, of (c)

t t. or :, o ir, n tl .

e

..~ 'r e !o 1

..

to mv •he for · 0=-

H,

in hich (c) i- th 00 fioi nt

in Q~o

ai'.

Th lu o (A) found to b =3.06 D0.584+o.a B::. 0•80

+

0.03, hioh f:! ve

DJ..282

.!:.5 4 0.85

~

+ _-0.2 - (0.03+ _D1.382 ) H

In th 2.0 _• ir t bove for . la ill no• o d for

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=

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(54)

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• D!SCH HGE TABLE FOR CIRCULAR WE!R •

Co pute F'ro the Form 11

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n

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