or,

v = 2gh.

(37)

In the case of a liquid whose motion is impeded by friction, the rate of flow is naturally less. The amount of reduction may be expressed by a fractional coefficient, attached to the preceding equation, denoting the proportion of head expended in overcoming the frictional resistance. Thus, the total head may be considered as divided into two portions, only one of which is available for producing velocity

The laws of fluid friction, which it will be useful to state at this point, differ materially from those relating to the surface-contact of solid bodies. They are as follows:

1. The friction is independent of the head, or pressure.

2. It varies directly as the area of the surface exposed to action.

3. It varies directly (or very

nearly so) as the square of the velocity. This, however, is only literally true so long as the rate of flow is sufficient to prevent the adherence of water to the surface in question.

Now, let us consider the case of

a horizontal culvert of length, x

Fig. 171.

(fig. 171), and sectional area, a, in which water is running full. Agreeably to the foregoing laws, we may express the amount of surface friction as

S = f.p.x. v2,

where is a coefficient to be determined later, and p is the perimeter of fluid section.

Now, assume the surface friction to be just counteracted by the difference of pressure upon the two faces of the length, x. That is

But this resultant pressure, (1

92) a, is due to a difference in head on each side of the culvert. Hence, we may substitute for it the expression for the pressure of the differential head-viz., wh1 a, in which w is

the weight of a cubic foot of water. At the same time, let R equation becomes

a

= and the

Xx v2

h1 = f.

Rw

v2

that we can replace it by the latter, without sensible error.

Whence,

2 g

This value for h1 determines the amount of head absorbed in overcoming friction. Its ratio to that given above (37) for simple discharge is expressed

x

R

by the coefficient: Ff The factor, f, varies with the nature of the surface of the conduit, and it is also found to depend, to a certain extent, on the relative diameter of the conduit and the rate of flow, being greater in small pipes than in large culverts, and at low velocities than at high speeds. Its value is found, however, to lie between 005 and 01, and 0075 may be taken as a serviceable mean for general use under normal conditions.

The symbol, R, standing for the area of fluid section divided by the perimeter, is referred to as the hydraulic mean radius, or the hydraulic mean depth. For circular and square culverts running full, and for circular culverts running half full, it is obviously equal to one-fourth of the diameter.

There are other sources of friction than that investigated above, and these cannot be overlooked in estimating the efficiency of the current issuing from a sluicing culvert :-

I. There is the friction due to the form of inlet at the reservoir. If an orifice in a thin plate, it has been found by experiment that

II. There is the friction at sudden enlargements or contractions of culvert area. Let the ratio in which the effective area is suddenly enlarged or contracted be designated r. Then, for abrupt enlargements,

and for abrupt contractions the same formula may be used, although the actual ratio of contraction is somewhat uncertain, being greater than the apparent ratio. The loss of head is due to the enlargement succeeding contraction.

are formulæ enunciated by Weisbach, r being the radius of curvature of the angle through which the culvert is bent

For

the centre line, and

The head necessary to overcome all these varied sources of friction must be deducted from the total head, and the residue will then represent the head available for producing velocity of exit, in accordance with the formula

where A is the area of opening, but in practice it is further necessary to take into account a modification due to the contraction of the free effluent leaving the culvert, by which the effective area of the current is less than the total area in a certain ratio, dependent on the shape of the outlet. This is brought about by the convergence of the particles into a vena contracta, or contracted vein.

Calling the pipe or culvert area unity, the following are coefficients (c) of actual discharge in the formula Q = c A v.

For wide openings, whose bottom is on a level with
that of the reservoir; for culverts with walls in a
line with the orifice,

.96.

For narrow openings, whose bottom is on a level with
that of the reservoir,

.86.

For sluices, without culverts or side walls,

.61.

In the foregoing investigation we have only credited the fluid current with the energy due to motion and to head or pressure, this being the case when the culvert is truly horizontal. When, however, there is a fall or inclination in the culvert the water possesses another source of energy, viz., energy of position, and this leads us to undertake an investigation into the principles which govern the flow of water in inclined pipes and culverts.

Reverting to the laws of fluid friction stated on p. 239, and remembering that when motion has become uniform, the acceleration and retardation of a current neutralise each other, we can form the following equation connecting the two. The acceleration is that due to the action of gravity on a body falling down an inclined plane of height, h, and length, l. Accordingly,

h

or,

substituting S for, the sine of slope, and introducing

so as to express the equation in terms of the hydraulic mean radius, we have

and this constitutes the basis of a very large number of expressions for the velocity, the values for C ranging from 70 to 100, according to the personal observation of different experimentalists.

Kutter's value for C, though complex, is recognised as the most generally reliable, and it is here given

in which a has the following numerical equivalents :—

009 for well-planed timber channel.

Strictly speaking, the amount of head introduced into the foregoing equation should be the total head reduced by that portion required to overcome the friction of entrance into the culvert, but when this latter is very small in comparison with the former, as it is in long conduits with moderate heads, the total head may be used without sensible error.

For the sake of example let us take the case of a horizontal culvert, 6 feet high by 4 feet wide, and find the amount of head required to produce an exit velocity of 4 feet per second. Assume a length of 100 feet, a squareedged entrance, and one bend of 60° in direction, with a radius of 5 feet. Then, by the preceding formulæ,

The head required to produce the same velocity through a simple sluice opening, as in a gate, will be as follows:

about one-half of the head required in the former case.

It may be interesting to compare the foregoing problems with a kindred one calculated by Kutter's formula. Suppose the culvert, as above, to have an inclination equal to that afforded by the head-viz., 64 inches-or, to simplify calculation, say 7 inches in 100 feet.

inch fall, is very

The difference, even allowing for the additional marked, but the results are not really comparable, being calculated on widely divergent lines from dissimilar conditions.

A very complete and interesting example of sluicing on an extensive scale is shown by the plan in fig. 173, which refers to the Canada tidal basin at Liverpool. * The main culverts are constructed partly in masonry and partly in iron. Those of iron are circular in section and lined with a layer of Portland cement inch thick, which is secured by dovetailed ribs or keys at close intervals along the castings. This work, although completed twenty years ago, is still sound and intact, exhibiting no signs of erosion or decay.

The centre of the basin is brought within the scope of the discharge by outlets in the floor of the northern portion, which is laid with concrete. The sluicing pipes are arranged in radiating lines beneath the floor (fig. 172), each being provided with a series of upper outlets along its length, and terminating in a splayed opening. To protect these openings heavy frames or discs of greenheart (fig. 174) are laid over them as covers, being secured by four strong links to foundation anchorages. When the sluices are not in use, these discs lie at rest upon their respective outlets, but under the pressure of flowing water within the culvert they are raised to the full extent allowed by the links, and the water rushes out in the form of annular jets, sweeping the circular area within its range.

This arrangement has been found extremely effective for the purpose * G. F. Lyster on "Dock Extensions at Liverpool," Min. Proc. Inst. C.E., vol. c.