THE ( lYIL PAGINLER AND ARCHITECTS JOLHNAL. 



[April, 



In determinirg tlie mcti(.n of w^ilpr issuii g from a rtn/ small orifice 

 in the bottom of a ryliiidrical vessel, it is clear tliat tlie tube may be 

 consideri'il of unequal but continuous boro, liow then can we find the 

 quantity discharged from the orifice? This we shall endeavour to do 

 approximately. We sliall first however seek for the maximum velo- 

 city of the issuing stream near the orifice. 



Suppose B B' the orifice, A A' a horizontal section of the fluid 

 above B B', tiiken at sucli a height above B B', that all the fluid be- 

 yond A A' may be considered at rest. L M the axis of the stream. 

 P any point in L M. Then the motion at P may be supposed wholly 

 vertical. Since L M is the axis of the stream, if then L P = £•, i' be 

 the velocity at P, the density of water 1, and g the measure of the 

 accelerating force of gravity, and/; the pressure at P, we shall have 

 dv dp r- _ 



Let now h be the depth of B B' below the surface of the water in the 

 vessel, the distance between B B' and A A'=: SA, then we have 



= —g{h — Sh}+C — ^, 

 because if ir = atmospheric pressure, t-|-j{A — 5A} = pressure at L 



.-.——n + gz—p + g {h — Sh}. 



So far as p is concerned, the velocity will be greatest when/) is least. 

 Let k be the value of ^ when/> has its least value, which is clearly t. 



k and d h being both extremely small. This expression becomes-^ = 



!) h. This, as far as it goes, apparently agrees with the method of 

 finding r, given in the books. We may remark, however, that in all 

 demonstrations we have seen, the great error is committed of esti- 

 mating the motion from the surface of the fluid, and assuming all par- 

 ticles in the same horizontal section to have the same vertical veloci- 

 ties. Now iu fact// becomes discontinious near the orifice, and wlien 

 the orifice is indefinitely small passes suddenly from />=? A -j-ir to 

 ;; = 7r, and consequently the equal number for r, which assume the con- 

 tinuity of/;, cannot be applied without further adaptation. 



To determine the velocity and quantity emitted at the orifice, re- 

 quires an altogether different kind of investigation. 



We shall here suppose that the tube is full, and that the fluid is 

 \erlically at rest within the vessel, even close to the opening; this, 

 although not strictly correct, will be found near enough to give toler- 

 ably accurate results. 



Let A := area of orifice. At time t from the commenceraent of 

 the motion, — suppose that if the jet had moved with a velocity in all 

 its parallel sections equal to its mean velocity of projection at BB', 

 it would have extended to a small distance, x, from B B'. At time 

 < -|-S/, let a: become ar-j-8x, then an additional quantity. At) 5. r, ha3 

 been sliot out from the orifice in the time it. Let R be the internal 

 force that efiected this; /I the pressure on the jet; — then, since the 

 only external force is Agh (neglecting Agx as extremely small), — 

 we have/) -\-R z=Agh; 

 and R 5< = A Sxv; 

 also, Ax5f)=/)5/= (AjA-R) St = Aghit—AtSj: 



,' . Ax — = A?/; — Ar -r-. When the motion is steady, — 

 ul dt ' 



dtJ . , dx 

 -=0,andr=-. 



, c' = jf A. 



c= V^yA. 



This is very near to the results of experiment, — If Q be the quantity 

 discharged iu time t, — 



Q = Ai V7a7 

 To determine the motion in a pipe of uniform bore : — 

 Suppose the tube inserted into a shallow nservoir of water kept 

 constantly full ; let the lube be straight, its diameter =: d, and length 

 = / ; let A =^ height of surface in reservoir above the point of eflBux. 

 When the motion is steady, let the mean vertical velocity of the 

 particles in the reservoir, just above the point where the tube enters, 

 be ^ times the velocity in the tube. Now, it is found that the resist- 

 ance of the tube varies as the square of the velocity, and that this 

 resistance arises fiom the inequalities of the interior of the tube. 

 If, therefore, I = length of tube, and d the diameter, the absolute 

 resistance will vary as i(/; but the mass of fluid varies as Id''. 



Let now x be the distance of any point of fluid in the pipe from 

 the point where the pipe enters the reservoir ; then, by the time that 

 X becomes x-\-tx, — a mass of fluid, rlid'', has had its velocity 

 changed from /i p to r. 



Therefore, if R measure the force which accomplishes this — B/ the 

 time of X becoming x -\-'h x — we shall have 



R5/= (I-m) r. irSxi^; 

 and, Tld'-lv ^ irghd'St— R5 t-e-rtdt', 

 e being a constant determined l)y experiment; 



.•.T^i^— = 'rj'A(i- — (1 — m)c — '"'■-e'^'''5'. 

 dt dl 



Therefore, when the motion is steady, and 



dv 

 dt 



1-M + e- 





1 + 



/ 



(i-rid 



If 1 -;t = -, or /* = g, and : 



57 



nearly, this becomes Eytel- 



wein's formula. 



If the water had first passed through a tube, length I and diameter 

 d, and then through a tube I', diameter d', we should have had 



gji 



l-M + ei + e. 



a' 



Evtelweiri's formula in inches is c' 



23iV;[TT 



57 d 



Example. — Water flows through a 9-inch main of 500U feet, and 

 then through a pipe of 4000 feet long and 5 inches diameter, the 

 height of head being 100 feet, what is the velocity of the discharge '. 



V = 23j 



v: 



^+ 57i ild' 



I = 12 X 5000 = 60000 

 r = 12 X 4000 = 48000 

 i = 12 X 100 = 1200 

 d = 9 d' = 5 



57 d — 513 57 i' = 285 



= 23i V 



1200 



1 + 



60UOO 



+ 



48000 



613 ■ 285 

 = 23i X V'4-2, nearly ; or 47 inches per second nearly. 

 We may remark that the value we have obtained for the mean ve- 

 locity of the discharge at a small orifice, VjT, is rather greater than 

 the velocity derived from experiment: this does not arise from any 



