Flow from Orifices

FLOW FROM ORIFICES If an opening be made in the side or base of a tank containing a fluid, convergent flow is set up from all sides towards the orifice. Owing to the inertia of the fluid the outer filaments of the issuing jet maintain their convergence for some little distance beyond the plane of the orifice. Up to this point the section of the jet grad ually diminishes. The section at which the jet first becomes paral The effective head producing efflux is now equal to the static head h plus the head equivalent to the kinetic energy of approach.

Flow over Weirs and Notches.

A weir may be looked upon as a large orifice whose upper edge extends to the surface of the water. If we assume that the velocity of efflux at a point distant x below the free surface is proportional to V 2gx, the discharge over such a weir is given by Q = KbH c.ft. per second, where b is the length of the weir in feet ; H is the head over the crest, measured to the level of still water above the weir; and K is an experimental coefficient, which varies with the type of weir and conditions of discharge.

In the case of a rectangular weir having a thin sharp edged crest and a vertical upstream face, the two most useful formulae are those of Francis and Bazin. In the Francis formula K = 3.33 while b is replaced by b—o.inH, where n is the number of end contractions. A weir with no end contractions is said to be "suppressed." In the Bazin formula, for a suppressed weir. These values of K apply where the area of the approach channel is so relatively large that the effect of the velocity of approach may be neglected. If, as is usually the where, in the Francis formula, Ii = ÷ 2g, v being the mean velocity in the approach channel, while in the Bazin formula P is the height of the weir crest above the bed of the channel.

The above formulae apply only to a weir having free access of air to the under side of the falling sheet or nappe. If the nappe clings to the crest or front face of the weir, or if free access of air is prevented, the discharge is increased.

Triangular Weirs.—If the weir is thin-crested and sharp edged, and if 0 be the angle between its two sides, Cippoletti Weir.—If the sides of a weir having two end con tractions be inclined outwards at an angle 0 with the vertical (fig. 6) the value of K in the formula Q = KbHi is sensibly in dependent of the head if 0 is such that the side slope is i hori zontal to 4 vertical. Such a weir is called a Cippoletti weir. The discharge is given by c.f.s.

if the velocity of flow in the approach channel is negligible, and by Q= 3'37b{ (H+h)/ — c.f.s.

as in the Francis formula, when the velocity of approach is taken into account.

Broad-crested Weirs.—Experiments indicate that if the width of the crest of a sharp edged weir is less than about •33H, the nappe will spring clear of the crest. Weirs with wider crests, in which the nappe adheres to the crest, are termed broad-crested weirs. Expressing the discharge over such a weir as Q=K'bH 3, values of K' have been determined experimentally for a large number of weir sections, and are given in any standard work on Hydraulics.

Precautions To Be Adopted in Weir Gaugings.

The stand ard sharp edged weir having a free discharge, or, for small quan tities, the right-angled triangular notch, are the only types for which the coefficients have been determined with sufficient accu racy to admit of use for measurement of flow without previous calibration.

For accurate measurement the following are essentials : I. Sharp edged weir sill, fixed so as to be incapable of vibra tion, having its face vertical and perpendicular to the direction of the stream, and, if rectangular, having its sill horizontal.

2. Clear discharge into air, with no adherence of the vein to the weir face.

3. Weir long in proportion to its depth, i.e., b>317.

4. H small in comparison with the depth of the approach chan nel, and sectional area of vein (bH) not greater than one-sixth that of this channel.

5. Suitable channel of approach. This should be as long and of as uniform section as possible so as to allow of the motion becom ing steady before reaching the weir. The length should, if pos sible, exceed 3oH, this ratio being increased where the length of weir is largely in excess of 3H.

6. Accurate determination of the head H. For accurate work the surface-level should not be taken in the stream itself, but in a stilling-box or pit from 18 in. to 2 ft. square communicating with the stream through a pipe of about i in. diameter. The zero of the gauge should be accurately adjusted to the level of the weir crest. For accurate work, where individual readings are to be taken, a hook gauge, provided with a vernier for reading to the nearest •00i ft., and with screw adjustment, is best.

Fluid Friction.

When fluid flows over a stationary surface a resistance to motion is experienced. At velocities so low as to give stream line flow this resistance is (r) directly proportional to the velocity of the fluid.

(2) directly proportional to the viscosity, µ.

(3) independent of the density, p.

(4) independent of the roughness of the surface.

At higher velocities at which the motion is sinuous or turbulent, the resistance is (I) proportional to v" where n usually lies between 1.8 and 2.0.

(2) varies as l21`-2, and is therefore independent of the viscosity when n= 2.

(3) varies as and therefore varies directly as the density when n = 2.

(4) increases with the roughness of the surface.

In the great majority of cases of practical importance, the motion is sinuous, and the frictional resistance may be written as R = fSv" where S is the wetted area in sq. feet, v is the velocity in ft. per sec., and f is a roughness coefficient whose value varies from about •002 for a smooth surface to about .004 for such a surface as that of a cast iron pipe after being in use for some years.

Disk Friction.

The resistance to the rotation of a disk in water is an example of fluid friction. It may be shown that this resistance is equal to n n+ rn+3 foot lb. 3 where r is the radius of the disk in feet and w is its angular ve locity. This resistance is of importance in the design of centri fugal pumps and turbines. In some high speed centrifugal pumps the energy expended in this way amounts to as much as 15% of the total energy given to the shaft.