V − v = 11.3 (r3/2/R) √i,
where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (Mém. de l’Académie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b = Ri/U2, where U is the mean velocity in the pipe; then V/U = 1 + 9.03 √b. A very useful result for practical purposes is that at 0.74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin.
| Fig. 84. |
§ 79. Influence of Temperature on the Flow through Pipes.—Very careful experiments on the flow through a pipe 0.1236 ft. in diameter and 25 ft. long, with water at different temperatures, have been made by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 11⁄2 in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v1.795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h = ζ (4L/D) (v2/2g), then for heads varying from 1 ft. to nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft. per second, the values of ζ were as follows:—
| Temp. F. | ζ | Temp. F. | ζ |
| 57 | .0044 to .0052 | 100 | .0039 to .0042 |
| 70 | .0042 to .0045 | 110 | .0037 to .0041 |
| 80 | .0041 to .0045 | 120 | .0037 to .0041 |
| 90 | .0040 to .0045 | 130 | .0035 to .0039 |
| 160 | .0035 to .0038 |
This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynolds’s equation is assumed h = mLVn/d3−n, and n is taken 1.795, then values of m at each temperature are practically constant—
| Temp. F. | m. | Temp. F. | m. |
| 57 | 0.000276 | 100 | 0.000244 |
| 70 | 0.000263 | 110 | 0.000235 |
| 80 | 0.000257 | 120 | 0.000229 |
| 90 | 0.000250 | 130 | 0.000225 |
| 160 | 0.000206 |
where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different temperatures Professor W. C. Unwin found that the resistance was proportional to constant × (1 − 0.0021t) and the values of m given above are expressed almost exactly by the relation
m = 0.000311 (1 − 0.00215 t).
In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3% of resistance for 10° F. increase of temperature.