Connection of Coefficients. Since F and C both express one fact, they must be in some way connected, and the connection is easily determined. For

so there is very little waste of head.

Discharge from a Short Pipe.-We have said that the values of the coefficients depend on the nature of the obstacles to flow; let us examine what takes place when a short pipe of length about 3d/2 is fitted outside the hole.

In the first place, the water issues in a full stream of area A, and therefore C, is unity.

In spite, however, of the gain in sectional area the discharge is found to be much less than for the simple orifice, CD being now only .815. It follows that C, is also .815, so there is a great falling-off in velocity. For the resistance we have

1.5

so that or of the head is now wasted. The reason of this increased resistance is shown at page 486.

Surface Friction. We have in the preceding case given a certain value to the head wasted, but we do not know exactly how it is wasted, whether from one cause only, or from more than one. We shall now proceed to consider certain causes which waste head separately, commencing with the most important.

In

Fig. 352 shows a thin plate with sharp edges completely immersed in water or other fluid, through which it is moving edgeways at V f.s. order to keep the plate moving at V f.s. it is found that an effort R is required, this effort balancing the friction of the water on the surface of the plate.

Fig. 352.

D

The value of R is determined by experiment, and it is found that

where

V2

R=fwS

2g

wweight of 1 c. ft. of the fluid
(including gaseous fluids),
S surface of the plate,

and ƒ is a constant depending only on the nature of the surface of the plate, and independent of the units employed. If we compare with the ordinary laws of friction, we find they are exactly opposite. For in the friction of dry surfaces R is-Ist, independent of V ; 2d, dependent on the pressure; 3d, independent of the surface; and each of these is exactly opposite in fluid friction.

The fact that R is independent of the pressure should be noticed, as it is a very common thing to meet statements to the effect that there is more loss by friction of a square foot of surface at the keel of a ship than there is at the water line. Experiment has conclusively shown that this is not so. The values of ƒ determined by Froude for a board 20 feet long, 19 inches deep, are

S is to be taken for the two sides, i.e. in the present case S = 20 × 12 × 19 × 2 sq. ins. The value of ƒ is found to be affected by the length of the board, ƒ being greater for short boards than for long, thus in the above table the first value for a board 2 ft. long is .0041, and for one 50 ft. long .0025. The effect is due to the fact that the first portion of the board drags the water along, and thus lessens the velocity of rubbing over the succeeding portion. These laws are of the first importance in the determination of the resistance of ships, but into this we cannot enter.

Surface Friction of Pipes.-The chief use we shall make of the preceding will be to determine the waste of head caused by surface friction of a pipe.

In Fig. 353 we show a horizontal pipe of uniform

A

B

Fig. 353.

A'

B'

diameter, through which fluid is flowing with velocity V.

Take two sections, AB, A'B', a distance x apart, and consider the motion of the water between

them. For clearness we may imagine two pistons at AB, A'B', which actually isolate this portion of the fluid. This portion of fluid being in uniform motion is balanced, and therefore

There is then a loss of pressure due to the friction; but we have seen pressure is equivalent to head, and a loss of pressure p-p' is equivalent to a waste of head (p-p)/w; hence if h' be the head wasted we have

S V2
h'=f .x.
·
A 2g

The quantity A/s is called the Hydraulic Mean Depth, and denoting it by m,

and denoting the length of the pipe now by 7, we obtain

1 V2 h' = 4ĥ 2g

as the total loss in the whole length of pipe.

The value of f is, as before stated, independent of the units, but looking at what we found as to the effect of length, we should be doubtful about what value to take for it. This difficulty we get over by taking the value of ƒ, not from Froude's results, but directly from experiments on pipes; we then are sure of having the correct value. It is found that 4f varies from .04 for I inch diameter to .02 for 4 inches and upward, varying, however, considerably according to the condition of the pipe. The values above are for a clean cast-iron pipe, and as an average value we use .03. In particular cases these values may be much exceeded. ·

Pressure in a Pipe. In the preceding we have used the term " 'pressure in the pipe," and have determined the loss of pressure. This being a question on which erroneous ideas are often expressed, it will be useful to see exactly what we mean by "the pressure in the pipe.” If a board be held in front of a jet of water issuing from a hose, a great pressure is felt on the board; or if the hand be held in a running stream a pressure is felt on it. Is it this pressure which we mean, and is it right to say, as it commonly is said, that the jet from the hose issues at high pressure?

To these questions we answer No! By the pressure in a pipe we mean the pressure of the portions of water on one another, not on a body which is held still so as to stop their flow. For example, in Fig. 353, the pressure of the water to the left of AB on that to the right is exerted on water which is moving away as fast as the pressing water follows it, and to feel this pressure we should, in the second example above, move the hand along with the stream. If we did this we know we should feel no pressure at all, there being no resultant pressure, but simply an equal pressure on back and front due to the depth below the surface, which would not be detected. In the case of the hose then the pressure in the jet is atmospheric simply, neither high nor low. We can also see this in another way; for, the pressure being equal in all directions, if the pressure in the jet were above the atmosphere, the outer portions would be thrown off radially, since the atmospheric pressure could not keep them together against the greater internal pressure.

Having now seen exactly what we mean by the pressure the question arises, How shall we measure it if required? Fig. 354 shows a pipe containing water. In the first case, suppose the water were still, the pressure could be measured by putting in a pipe of any shape as AB or CD with an open end; and the water