assume the same velocity. The laws of this lateral communication of motion, or internal friction, of fluids, are not known exactly; but its effects are known thus far :-that the energy due to differences of velocity, which it causes to disappear, is replaced by heat in the proportion of one thermal unit of Fahrenheit's scale for 772 foot pounds of energy, and that it causes the friction of a stream against its channel to take effect, not merely in retarding the film of fluid which is immediately in contact with the sides of the channel, but in retarding the whole stream, so as to reduce its motion to one approximating to a motion in plane layers perpendicular to the axis of the channel (Article 625).

640. Friction in an Uniform Stream.-It is this last fact which renders possible the existence of an open stream of uniform section, velocity, and declivity. In hydraulic calculations respecting the resistance of this, or any other stream, the value given to the velocity is its mean value throughout a given cross-section of the stream A,

..(1.)

The greatest velocity in each cross-section of a stream takes place at the point most distant from the rubbing surface of the channel. Its ratio to the mean velocity is given by the following empirical formula of Prony, where V is the greatest velocity in feet per second :

In an uniform stream, the dynamic head which would otherwise have been expended in producing increase of actual energy, is wholly expended in overcoming friction. Consider a portion of the stream whose length is 1, and fall z. The loss of head is equal to the fall of the surface of the stream, according to Article 623; and the expenditure of potential energy in a second is accordingly

zęQ=zęvA.

Equating this to the work performed in a second in overcoming friction, viz., v R, we find

or dividing by common factors, and by the area of section A, we find for the value of the fall in terms of the velocity

STREAMS-HYDRAULIC MEAN DEPTH.

587

Let s be what is called the wetted perimeter of the cross-section of the stream; that is, the cross-section of the rubbing surface of the stream and channel; then

S = 18;

and dividing both sides of equation 3 by 7, we find for the relation between the rate of declivity and the velocity,

A

8

"of the stream;

Iis what is called the "HYDRAULIC MEAN DEPTH and as the friction is inversely proportional to it, it is evident that the figure of cross-section of channel which gives the least friction is that whose hydraulic mean depth is greatest, viz., a semicircle. When the stability of the material limits the side-slope of the channel to a certain angle, Mr. Neville has shown that the figure of least friction consists of a pair of straight side-slopes of the given inclination connected at the bottom by an arc of a circle whose radius is the depth of liquid in the middle of the channel; or, if a flat bottom be necessary, by a horizontal line touching that arc. For such a channel, the hydraulic mean depth is half of the depth of liquid in the middle of the channel.

641. Varying Stream.-In a stream whose area of cross-section varies, and in which, consequently, the mean velocity varies at different cross-sections, the loss of dynamic head is the sum of that expended in overcoming friction, and of that expended in producing increased velocity, when the velocity increases, or the difference of those two quantities when the velocity diminishes, which difference may be positive or negative, and may represent either a loss or a gain of head. The following method of representing this principle symbolically is the most con

venient for practical purposes. In fig. 253, let the origin of coordinates be taken at a point O completely below the part of the stream to be considered; let horizontal abscissæ x be measured against the direction of flow, and vertical ordinates to the surface of the stream, z, up

wards. Consider any indefinitely short portion of the stream whose horizontal length is dx; in practice this may almost always be considered as equal to the actual length. The fall in that portion of

the stream is dz, and the acceleration - dv, because of v being opposite to x. Then modifying the expression for the loss of head due to friction in equation 3 of Article 640 to meet the present case, and adding the loss of head due to acceleration, we find

In applying this differential equation to the solution of any particular problem, for v is to be put Q÷A, and for A and 8 are to be put their values in terms of x and z. Thus is obtained a differential equation between x and z, and the constant quantity Q, the flow per second. If Q is known, then it is sufficient to know the value of z for one particular value of x, in order to be able to determine the integral equation between ≈ and x. If Q is unknown, the dz values of z for two particular values of x, or of z and (the dx

declivity), for one particular value of x, are required for the solution, which comprehends the determination of the value of Q.

642. The Friction in a Pipe Running Full produces loss of dynamic head according to the same law with the friction in a channel, except that the dynamic head is now the sum of the elevation of the pipe above a given level, and of the height due to the pressure within it. The differential equation which expresses this is as follows:-Let dl be the length of an indefinitely short portion of a pipe measured in the direction of flow, s its internal circumference, A its area of section, z its elevation above a given level, p the pressure within it, h the dynamic head. Then the loss of head is

-dh = -dz

The ratio

dh dl

is called the virtual or hydraulic declivity, being the

rate of declivity of an open channel of the same flow, area, and hydraulic mean depth. This may differ to any extent from the

actual declivity of the pipe,

dz dl

=

0, and the first term

When the pipe is of uniform section, dv of the right-hand side of equation 1 vanishes. When the section of the pipe varies, s and A are given functions of l. If Q is given, v = Q A is also a given function of l; and to solve the equation completely, there is only required in addition the value of h for one particular value of l. If Q is unknown, the dh for one αι

values of h for two particular values of 1, or of h and

FLOW IN PIPES-SUDDEN ENLARGEMENT.

589

particular value of 1, are required for the solution, which comprehends the determination of Q.

643. Resistance of Mouthpieces.-A mouthpiece is the part of a channel or pipe immediately adjoining a reservoir. The internal friction of the fluid on entering a mouthpiece causes a loss of head equal to the height due to the velocity multiplied by a constant depending on the figure of the mouthpiece, whose values for certain figures have been found empirically; that is to say, let - Ah be the loss of head; then

f being a constant.

For the mouthpiece of a cylindrical pipe, issuing from the flat side of a reservoir, and making the angle i with a normal to the side of the reservoir, according to Weisbach,

ƒ = 0·505 + 0.303 sin i + 0.226 sin' i. ..... (2.)

.........

644. The Resistance of Curves and Knees in pipes causes a loss of head equal to the height due to the velocity multiplied by a coefficient, whose values, according to Weisbach, are given by the following formula:-For curves, let i be the arc to radius unity, r the radius of curvature of the centre line of the pipe, and d its diameter.

For knees, or sudden bends, let i be the angle made by the two portions of the pipe at either side of the knee with each other; then

ƒ“ = 0·9457 sin2 1

i

+ 2·047 sin1

(2.)

645. A Sudden Enlargement of the channel in which a stream of liquid flows, causes a sudden diminution of the mean velocity in the same proportion as that in which the area of section is increased. Thus, let v, be the velocity in the narrower portion of the channel, and let m be the number expressing the ratio in which the channel is suddenly enlarged: the velocity in the enlarged part

is

V1 Now it appears from experiment, that the actual energy

m

due to the velocity of the narrow stream relatively to the wide

stream, that is, to the difference v1

is expended in over

coming the internal fluid friction of eddies, and so producing heat; so that there is a loss of total head, represented by

646. The General Problem of the flow of a stream with friction

v

is thus expressed :—Let hand h + be the total heads

2g'

at the beginning and end of the stream respectively; then the loss of total head is represented by

where the right-hand side of the equation represents the sum of all the losses of head due to the friction in various parts of the channel. (See p. 647.)

SECTION 4.-Flow of Gases with Friction.

647. The General Law of the friction of gases is the same with that of the friction of liquids as expressed by equation 1, Article 638, the value of the co-efficient ƒ being

0.006, nearly,

for friction against the sides of the pipe or channel. For a cylindrical mouthpiece, the co-efficient of resistance is 0.83; for a conical mouthpiece diminishing from the reservoir, 0.38.

1

10

When the pressures at the beginning and end of a stream of gas do not differ by more than of their mean amount, problems respecting its flow may be solved approximately by means of the above data, treating it as if it were a liquid of the density due to the lesser pressure, as in the approximate equation of Article 637.

In seeking the exact solution of the flow of a gas with friction, it is necessary to take into account the effect of the friction in producing heat, and so raising the temperature of the gas above what it would be if there were no friction, as supposed in Section 2. In the flow of a perfect gas with friction, if the heat produced by the friction is not lost by conduction, the friction causes no loss of total