For the smoother surfaces the friction varied as the 1.85th power of the velocity. For the rougher surfaces the power of the velocity to which the resistance was proportional varied from 1.9 to 2.1. This is in agreement with Froude’s results.
Experiments with a bright brass disk showed that the friction decreased with increase of temperature. The diminution between 41° and 130° F. amounted to 18%. In the general equation M = cNn for any given disk,
ct = 0.1328 (1 − 0.0021t),
where ct is the value of c for a bright brass disk 0.85 ft. in diameter at a temperature t° F.
The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of the results obtained with the disks and Froude’s results on planks 50 ft. long. The values given are the resistances per square foot at 10 ft. per sec.
| Froude’s Experiments. | Disk Experiments. | ||
| Tinfoil surface | 0.232 | Bright brass | 0.202 to 0.229 |
| Varnish | 0.226 | Varnish | 0.220 to 0.233 |
| Fine sand | 0.337 | Fine sand | 0.339 |
| Medium sand | 0.456 | Very coarse sand | 0.587 to 0.715 |
VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.
§ 71. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.
| Fig. 80. |
In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction must be equal. Let fig. 80 represent a very short portion of the pipe, of length dl, between cross sections at z and z + dz ft. above any horizontal datum line xx, the pressures at the cross sections being p and p + dp ℔ per square foot. Further, let Q be the volume of flow or discharge of the pipe per second, Ω the area of a normal cross section, and χ the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ ℔, and fall through a height −dz ft. The work done by gravity is then