eddying motions being set up in it. Actually, however, what happens is, that on opening into the large pipe the water breaks away in all directions; there is a sort of main stream flowing on, but the corners are full of broken water which is continually joining the main stream, and its place being supplied from the small stream entering. The water then flowing across the section BB has all kinds of cross motions, and head is wasted in producing these motions.

The effect is due to a stream moving with velocity v impinging on a larger stream moving at velocity V, the relative velocity or velocity of striking being - V.

We will then first examine a somewhat simpler case. Fig. 360 represents a bucket held stationary, the sectional area of the bucket being A ;

a stream of sectional area a enters the bucket at a velocity - V and strikes the bottom, being thus entirely stopped, and dropping down vertically when it pours out as more water enters. The head originally used to produce the velocity V is thus wholly wasted, so the waste of head is (v-V)2/2g, nothing but confused motions being finally produced by it.

Fig. 360.

Next, let the whole system move on with velocity V, the bucket has now the velocity V and the stream a velocity ; the velocity of striking is not altered, and hence we infer that the amount of confused motion produced is the same, the water moving on also as a whole at the velocity V of the bucket. But the waste of head is due to the production of the confused motion, and we conclude that it will, therefore, as before, be (V)2/2g.

The next step is obvious; we have only to replace the wooden bottom of the bucket by the water surface at BB, and we have the present case. We reason that this cannot affect the loss of head, it being immaterial

whether the surface struck be of water or of wood, and we say then finally

We have here a verification of the statement on page 469 regarding hydraulic resistances, for we will now express the loss by means of a coefficient of resistance.

We have two distinct velocities of flow, v and V, either of which may be selected, so that F, the coefficient, will have two values. We write then

In either case F is a constant, depending only on the nature of the source of the resistance, as stated in the general law. We must be careful to connect together the proper coefficient and velocity, to remember which take note that the larger coefficient goes with the smaller velocity, and vice versa.

Sudden Contraction.-In Fig. 361 we have the

с B

C B

A

Fig. 361.

reverse case to that just considered. The stream now contracts to CC and then expands again to BB, so that there are two distinct actions to consider.

In the first of these little waste takes place, because little broken water is caused, the water moving quietly round the corners AA and being kept together

by the surface of the pipe. There will be a little loss, such as we have seen to occur in the case of a simple orifice.

But in the second action there will be considerable waste, this being a sudden expansion similar to that just considered, and consequently the waste being expressed in the manner just discussed. We neglect then the first part because the loss is so small, and we have

F = (m − 1)2,

where m is the ratio of a to the contracted section at C, and the coefficient refers to v2/2g.

The value of m will vary with different ratios of A to a, and is believed to be approximately given by the empirical formula

We see here the explanation of the increased waste of head caused by fitting a short pipe to an orifice (page 470), that being simply a case of the preceding, in which A is indefinitely large compared with a. We can calculate the amount thus. Let

The remaining part of the .505 is due to the friction of the corners of the orifice and of the pipe.

The preceding results and others which we cannot examine into are collected in the following table, the values of F being determined by experiment.

When water flows through a channel of varying section, containing several causes of resistance, the waste of head at each will be given as a multiple of 2/2g, where is the velocity past that particular obstacle. It is usual, for convenience, to express the

whole waste in terms of one selected velocity; suppose the velocity selected be V, then we have

where F' is a new coefficient derived from F by multiplication by v2/V2, or if a and A be the sectional areas at the parts considered, by multiplication equally by A2/a2. F' is called the coefficient of resistance referred to the velocity V. In this way values F', F", etc., are obtained for all the obstacles, and then we have

ΣF being called the total coefficient referred to the velocity V.

Flow of Gases under Small Differences of Pressure. When a gas flows, the density-i.e. w varies as the pressure varies, and also varies with alterations of temperature. The flow then generally becomes a question of Thermodynamics.

If, however, the differences of pressure be small—that is, as is often the case in practice, such as are measured by a few inches of water--and no heat be supplied, the gas flows practically as a liquid having the same mean density, and we will examine this case. Let

T=absolute temperature of gas,

then T461+t, where t is the Fahrenheit temperature, V = volume of 1 lb. in c. ft.,

AP the small difference of pressure producing

the flow in lbs. per sq. ft.