179.

Motion of water in pipes.

The circumstances of the motion of water in pipes,

are closely analagous to those of its motion in open channels. The forces which tend to impart motion

E

R

D

C

C B

Fig. 157.

are dependent upon the weight of the water in the pipe, and upon the height of the water in the upper reservoir. Those which tend to prevent motion depend upon the depth of water in the lower reservoir, friction in the pipe, adhesion, and shocks arising from irregularities in the bore of the pipe. The retardation due to shocks will, for the present, be neglected.

Let AB represent a straight cylindrical pipe, connecting two reservoirs R and R'. Suppose the water to maintain its level at E, in the upper, and at C, in the lower reservoir. Denote AE, by h, and BC, by h'. Denote the length of the pipe, by 7, its circumference, by c, its cross-section, by a, its inclination, by p, and the weight of a unit of volume. of water, by w.

Experience shows that, under the circumstances above indicated, the flow soon becomes uniform. We may then regard the entire mass of fluid in the pipe as a coherent solid, moving with a mean uniform velocity down the inclined plane AB.

The weight of the water in the pipe will be equal to wal. If we resolve this weight into two components, one perpendicular to, and the other coinciding with the axis of the tube, we shall have for the latter component, wal sing. But I sing is equal to DB. Denoting this distance by h", we shall have for the pressure in the direction of the axis, due to the weight of the water in the pipe, the expression wah". This pressure acts from A towards B. The pressure due to the weight of the water in R, and acting in the same direction, is wah.

The forces acting from B towards A, are, first, that due

to the weight of the water in R', which is equal to wah'; and, secondly, the resistance due to friction and adhesion. This resistance depends upon the length of the pipe, its circumference and the velocity. It has been shown, by experiment, that this force may be expressed by the term,

cl(kv + k'v2).

Since the velocity has been supposed uniform, the forces acting in the direction of the axis, must be in equilibrium. Hence,

wah+wah" = wah' + cl(kv + k'v2);

a

The factor is equal to one-fourth of the diameter of the

The values of m and n, as determined experimentally by PRONY, are,

m = 0.00017, and

Hence, by substitution,

n0.000106.

.00017v+.000106v2 = ds.

If v is not very small, the first term may be neglected, which will give,

v = 48.56√√ds.

If we denote the quantity of water delivered in n seconds, by Q, we shall have,

The velocity will be greatly diminished, if the tube is curved to any considerable extent, or if its diameter is not uniform throughout. It is not intended to enter into a discussion of these cases; their complete development would require more space than has been allotted to this branch of Mechanics.

General Remarks on the distribution and flow of water in pipes.

180. Whenever an obstacle occurs in the course of an open channel or pipe, a change of velocity must take place. In passing the obstacle, the velocity of the water will increase, and then, impinging upon that which has already passed, a shock will take place. This shock consumes a certain amount of living force, and thus diminishes the velocity of the stream. All obstacles should be avoided; or, if any are unavoidable, the stream should be diminished, and again enlarged gradually, so as to avoid, as much as possible, the necessary shock incident to sudden changes of velocity.

For a like reason, when a branch enters the main channel, it should be made to enter as nearly in the direction of the current as possible.

All changes of direction give rise to mutual impacts amongst the particles, and the more, as the change is more abrupt. Hence, when a change of direction is necessary, the straight branches should be made tangential to the curved portion.

The entrance to, and outlet from a pipe or channel, should be enlarged, in order to diminish, as much as possible, the coefficients of ingress and egress.

When a pipe passes over uneven ground, sometimes ascending, and sometimes descending, there is a tendency to a collection of bubbles of air, at the highest points, which

may finally come to act as an impeding cause to the flow. There should, therefore, be suitable pipes inserted at the highest points, to permit the confined air to escape.

Finally, attention should be given to the form of the crosssection of the channel. If the channel is a pipe, it should be made cylindrical. If it is a canal or open aqueduct, that form should be given to the perimeter which would give the greatest cross-section, and, at the same time, conform to the necessary conditions of the structure. The perimeter in open channels is generally trapezoidal, from the necessity of the case; and it should be remembered, that the nearer the form approaches a semi-circle, the greater will be the flow.

Capillary Phenomena.

181. When a liquid is in equilibrium, under the action of its own weight, it has been shown that its upper surface is level. It is observed, however, in the neighborhood of solid bodies, such as the walls of a containing vessel, that the surface is sometimes elevated, and sometimes depressed, according to the nature of the liquid and solid in contact. These elevations and depressions result from the action of molecular forces, exerted between the particles of the liquid. and solid which are in contact; from the fact that they are more apparent in the case of small tubes, of the diameter of a hair, these phenomena have been called capillary phenomena, and the forces giving rise to them, capillary forces.

These forces only produce sensible effects at extremely small distances. CLAIRAUT has shown, that when the intensity of the force of attraction of the particles of the solid for those of the liquid, exceeds one-half that of the particles of the liquid for each other, the liquid will be elevated about the solid; when less, it will be depressed; when equal, it will neither be elevated nor depressed. In the first case, the resultant of the capillary forces is a force of capillary attrac tion; in the second case, it is a force of capillary repulsion ; and in the third case, the capillary forces are in equilibrium. The following are some of the observed effects of capillary

action: When a solid is plunged into a liquid which is capable of moistening it, as when wood or glass is plunged into water, the surface of the liquid is heaped up about the solid, taking a concave form, as shown in Fig. 158. When a solid is plunged into a liquid which is not capable of moistening it, as when glass is plunged into mercury, the surface of the liquid is depressed about the solid, taking a convex form, as shown in Fig. 159. The surface of the liquid in the neighborhood of the bounding surfaces of the containing vessel takes the form of concavity or convexity, according as the material of the vessel is capable of being moistened, or not, by the liquid.

་

Fig. 158.

Fig. 159.

These phenomena become more apparent when, instead of a solid body, we plunge a tube into a liquid, according as the material of the tube is, or is not, capable of being moistened by the liquid, the liquid will rise in the tube or be depressed in it. When the liquid rises in the tube, its upper surface takes a concave shape; when it is depressed, it takes a convex form. The elevations or depressions increase as the diameter of the tube diminishes.

Elevation and Depression between plates.

182. If two plates of any substance are placed parallel to each other, it is found that the laws of ascent and descent of the liquid into which they are plunged, are essentially the same as for tubes. For example: if two plates of glass parallel to each other, and pretty close together, are plunged into water, it is found that the water will rise between them to a height which is inversely proportional to their distance apart; and further, that this height is equal to half the height to which water would rise in a glass tube whose internal diameter is equal to the distance between the plates.