^mm ~ f^^^K^ C UC-NRLF ^C DIAGRAMS FOR THE SOLUTION OF THE KUTTER AND BAZIN FORMULAE FOR THE FLOW OF WATER When any three of the four variables, Velocity. Slope, Hydraulic Radius, and Roughness, are known, the fourth can be read off at once, in English or metric units, without using a straight edge. PREPARED BY KARL R. KENNISON, M. ME. Soc. C. E. 815 Grosvenor Building PROVIDENCE, R. I. 1913 PRICE $1.00 /v \ THE KUTTER AND BAZIN FORMULAE In the absence of actual discharge measurements, which should always be preferred to the best computations, the formulae represented in these diagrams are commonly relied on to compute the velocity of water flowing in open channels and pressure conduits. The Kutter formula is generally preferred to the Bazin formula, for all classes of channels, especially hi computations of river flow. The Bazin formula is not generally applied to channels over twenty feet wide. These formulae assume a condition of uniform flow, and depend for accurate results on the right choice of a coefficient of roughness to fit the channel hi question. Reference should be had to the many published works on Hydraulics for a discussion of the proper application of the formulae, the measurement of surface slope, the determination of the coefficient of roughness, and the effects of bends and irregularities in the channel bed, which practically increase the coefficient of roughness. Uncertainties in the application of the formulae do not warrant a more precise solution than can be obtained easily with these diagrams. NOTATION V....Mean velocity of water in uniform motion, in feet per second. (The marginal scale of velocities is in meters per second.) s.... Slope of free water surface or hydraulic gradient, friction head -r length. S....1000 x s, or slope in feet per thousand feet (or meters per thousand meters). R. ..Hydraulic radius, or sectional area of stream-i-wet perimeter, in feet. (The marginal scale of hyd. radii is in meters.) For ordinary river beds, R practically the mean depth. n.... Coefficient of roughness in Kutter formula. y.... Coefficient of roughness in Bazin formula. Some of the values of n and y in common use are shown below the diagrams. They are average values and should be varied to suit the condition of the surface in question: e. g., for planed boards well laid and with smooth end joints n is commonly assumed =.009 instead of .010: For concrete lined tunnels, where only ordinary care is taken to obtain a smooth interior and where the obstruction due to vegetable growths must be anticipated, n should be assumed=.013 or .014 instead of .010, the value given for smooth cement: Swollen rivers encumbered with detritus might require a value of n as high as .045, or even higher in torrents spending part of their energy in rolling boulders along and across the bottom. 288611 EXAMPLES ILLUSTRATING USE OF DIAGRAMS (1) To find the carrying capacity of a circular tunnel 8 ft. in dia., lined with brick- work in poor condition, flowing full under a hydraulic gradient of 5 feet per mile. The roughness of this lining is about equal to that of rubble masonry, or say n=.017. R=2 ft. S=.95 ft. per thousand ft. Area of tunnel=50.3 sq. ft. A vertical line up from the intersection of n=.017 and R=2 intersects S=. 95 at V=4.3. Therefore the carrying capacity 4.3 ft. per sec. x 50.3 sq. ft.=216 cu. ft. per sec. Practically the same result would have been obtained by the Bazin diagram, using y=.46. (2) The following simultaneous stream measurements were made covering a certain river length: Di-charge=3340 cu. ft. per sec., S=2.35 ft. per thousand ft., mean V=5.2 ft. per sec., mean R=4.0 ft. From the computed coefficient of roughness, which under the conditions of this problem we may assume constant, find the slope along this stretch of the river, when the same quantity is flowing, obstructed by a dam at some point down- stream which raises the water surface so that mean R 12.5 ft. and mean sectional area of stream=2350 sq. ft. A vertical line down from the intersection of V=5.2 and S=2.35 intersects R=4.0 at n=.035. After the dam is built, the velocity is 3340 cu. ft. per sec.-r-2350 sq. ft.= 1.42 ft. per sec. A vertical line up from the intersection of n=.03o and R=12.5 intersects V=1.42 at a point lying in the group of lines for S=.03 but below the one corresponding to n=.035. Interpolating therefore between the lines S=.01 and S=.03, using in each case the one corresponding to n=.035, we find that our point lies on S=.025, which is the required slope in ft. per thousand ft. due friction V\eai This last example illustrates the application of the diagrams to the problem of "back- flowage" above dams and other obstructions. The intersection of V (Mean Velocity in Ft. per sec.) and S (Slope in ft. per thousand ft.) is vertically over the intersection of n (Coefficient of Roughness and R (Hydraulic Radius in ft.) Note that S in diagram 1000 X s in Formula V= 1.2- v= For Metric Units 7 o . I . .00155 " + ^rr + s~~ -.6=R Diagram FLOW OF WATER Formula of Ganguillet and Kutter commonly called the KUTTER_FORMULA By Kar~l R. Kennison C-Opyright 1312. For English Units 41.6 Some values of n in common use: * Planed boards or smooth cement, .010; Well laid brickwork, .013; Rubble masonry, .017; Very firm gravel, .020; Earthen canals in good order, .025; Ordinary earthen river beds with occasional stones and weeds, .030; Earthen river beds in bad order, .035 or more. The intersection of V (Mean Velocity m ft- per se and S (Slope in ft. per thousanc is vertically over the intersection oF Y (Coefficient of Roughness) and R (Hydraulic Radios in ft.) V. vJ C.) </> ft) s : -3 3 2 -2 -Z X X X- 'xiv X X X x 10 X 1 i= ' J x." , J 0- -L < * X > X " X ' x" X* X* x X ' ' X x 7 x x x> O 1 X x / ' x x ' X x" x j x x x _i_3 IX x x7k p X 7 X X f X X x X X x ' 1 7 7 X jx" X 1? ^x S X X x ^ X / x fX '" x X 7 x x x x x /\ 5 -4.5 4 1 "rO X:x X > X X x x X . xlx . X X x / ~ X ] X 1 ^ \A x^ X 7 X x-^- x X ' ^ X ^ x x- x x' X X ' Note that S ir\ diagra 1000 x S in Formula X -' X X X ^ 'X X X ? x x X /^ g X X x x / X x X / x X x X 7 x 7 x ' i"x X X X x x' x" X s . x x x X x x ^ x x x x x / x 1 x x X X x*' ,< X ^ ' x X ^ '' x -2** 1 Z ""* ,x X x x X / x' x- x X 7 X ^ X ^'L $ X x x' x' x x X X x 7 7 X X X x" Px 1 x x x x^ x' X ' x x x X x X X X X s x - x / i X ''X x X x x X X / x x' X 7 'X ' X X x x X x x x^ ^x X x x x X x X x X x X x X x x x x x _c x" x' ^ X x' X X ^ S x X x x X x x ' 1*8 in 1.6 -. b x" 7 X ^ r XX X x x x" / x 1.5 1 X x . x' ,x' x X" X x X X X X x^ x X X x X X x ~ x , '' x x' ' X x ^X X x- x ^ -' x ^ x ( X -1.2 -1 ? i ___ ^\^ x X X 1 x x' x X > X , x x X, I 3,5 X x X g x x xl ^ .X / x <^ ? x xX _/ X x X x x x / x x ^ * x x 1 ^ x . X x *j5>- x ^ ' x .9 .8- x x X 3?., x x ( 7 X X ^ X X _XO 1 X X ,x x x x x x ? x X .8 ' X X > >/^ X X X x; X ' ^ ' x ^x !.3 1 x X X x" X X x X X X ' x ^ x ' _ * x 1 X 7 S x x ^ * X" ^ "^ ^ X x * s o x -' J 1 x- ^ X X X X " x -.6 .6 V x x x x ,.x' X LX x X & x -' ' .6 D .5V (0 c 9- R X x ? XI 3 ."^x X x x '' s x , x X' x 1 X '^ ^ ^1 - X ,x- ^-' ^^ . X ' ^ '' X 7 x x x /s* .x X| / x x , X X X xX" X ^ x "*" / ^x X' X ' . 'S ^ 1 x ro ' c X , X X x X fX ^ x J* x . .X x '> X X X- x x' x ^x x" X ^ X X ^. ^x** , x- x X x X s X " ,x ^ X X ^x X X ^ T-+*^ / Ufl o x x ^ - X x* ' < X X X X 1 ^ ^^ ^x X ^,. _^X H=3 io .26- f . [X J X a . X X X x' *s ^ _ ci:*" 1 ; XT x ^ ^ , x r ,*' x J x / / / ' .23- .2 x ^ " X X x'' x . ^ x x'- X x" .U 7 ^x^. X "" ^ -' * . x' pX X ?2 I C 3x x / 8 _e x x' X X X X / ^ > /' x ' x x _r X X X x X" X X <" .16- x jS X x" X X X .- > ^ ^ X ? 15 .4 x- x X - X^ "^ x / / / ~~6 c ' x 1* X x " .. x ^x -X .x-' .-' X X" ^ ,- X x > x / ' / / UJ "*" 17 X X ,- X ^ X -' X X y r r x- '"" 5 x x x^ / / / f X --' x ^ X X" X x ' x- X x' ? / ' \/ x VMX ? / ^ x X x x x- ' x x ^ X ^ ^ ^^ -^ ^ x' r.35^x ^x- -4.5^0 5 08- /s x^X x' X x < 2 xl^x V2 -.2 5* x" X / / ^ / / / P' X 1 / -3 ^ .06- X X x^ X x- x^ -- 3^ ; L. ! s J X x / x / X ,' 7 V* / '/\ / ~ / ^ J ^X x 1 X X, X 5 X X ^ 'G~? i .it -. K -.1' X x x X X X Q - / / / ' r- Od ^ X x- X ^ ^!x 5^ ^- X i X x ' x / , '~ / u / 2.6 E i S X X / X ,x ' / ' L^^'7 x' / V=.03- j: (S\ Tin uJ x x- x !x^ ,.. x' ^x" * . ! i ~t.J X X X -F la 3 2. 2. -1. x" !/ X x / / .- / / c / X 2 . x S X J 7 / / ' / '*/ 1.8 u !.&" KV ' ' / ^ X x 7 V x 7 / / ** .5 145" ?>^ X J .- x x x ' /: x x'X' 1 V X --' 1 ( 7 ^ x Z / .5 > 1.2. 3 X ' tX X yp . ? x X / * f A U- ^ x X x ' Jx 7 / x / z -* _ ^. h :.G ..3 .5 o 7 - lX x X x ; , X A* 3 X / / .5 ~ s ^ X ' x X X <^' , ^ / R / X xiX ' X X x X"* C 3 , 7 / -' x^ x X *4 Z 4. x x , !. I. "l X / X ^x X X X . r j/. x - x X x- 0/ , x x x 7 ^ Diagram of FLOW OF WATER By the BAZIN FORMULA c *4 ! X X^ X x x x X X / flj- X x^ X' ^ x i X x ' /" / x X x . x x 2 x xl / X x x X"^ x -?x , x - X / X x X IX -- ' x X x / *-.!6- ^ x x . X X X / x' ^ 5 x X x- X ,x ( J ," x x X X By Karl R. Kennison u .1Z .1- X X i X x ^ 1 35 ^ x ' - XT x X ' X x x X' / For V- Metric 87 Uni \ ; "R" -l-s 5 i For v= (-righ-*- 1913 English Units 87 -VrTT Qfl x X X" x .o - .' x~ X 2 x X X x x X X x-^ x 05 x 1 ' x x -t j ,xl x X , .18 > .04- R=.035- / x X X A'\^ x - * X X x' X =! j M 1 r\o X X / Some values of y in common use: Planed boards or smooth cement, .06; Well laid brickwork, .16; Rubble masonry, .46; Very firm gravel. .80; Earthen canals in good order, 1.30; in bad order, 1.75. For computing river flow the Kutter formula is generally considered more accurate than Baiin. Kemington Press, Providence UNIVERSITY OF CALIFORNIA LIBRARY I