 |
| Fig. 83. |
In 1886, Professor W. C. Unwin plotted logarithmically all the most trustworthy experiments on flow in pipes then available.[1] Fig. 83 gives one such plotting. The results of measuring the slopes of the lines drawn through the plotted points are given in the table.
It will be seen that the values of the index n range from 1.72 for the smoothest and cleanest surface, to 2.00 for the roughest. The numbers after the brackets are rounded off numbers.
The value of n having been thus determined, values of m/dx were next found and averaged for each pipe. These were again plotted logarithmically in order to find a value for x. The lines were not very regular, but in all cases the slope was greater than 1 to 1, so that the value of x must be greater than unity. The following table gives the results and a comparison of the value of x and Reynolds’s value 3−n.
| Kind of Pipe. | n | 3−n | x |
| Tin plate | 1.72 | 1.28 | 1.100 |
| Wrought iron (Smith) | 1.75 | 1.25 | 1.210 |
| Asphalted pipes | 1.85 | 1.15 | 1.127 |
| Wrought iron (Darcy) | 1.87 | 1.13 | 1.680 |
| Riveted wrought iron | 1.87 | 1.13 | 1.390 |
| New cast iron | 1.95 | 1.05 | 1.168 |
| Cleaned cast iron | 2.00 | 1.00 | 1.168 |
| Incrusted cast iron | 2.00 | 1.00 | 1.160 |
With the exception of the anomalous values for Darcy’s wrought-iron pipes, there is no great discrepancy between the values of x and 3−n, but there is no appearance of relation in the two quantities. For the present it appears preferable to assume that x is independent of n.
It is now possible to obtain values of the third constant m, using the values found for n and x. The following table gives the results, the values of m being for metric measures.
Here, considering the great range of diameters and velocities in the experiments, the constancy of m is very satisfactorily close. The asphalted pipes give rather variable values. But, as some of these were new and some old, the variation is, perhaps, not surprising. The incrusted pipes give a value of m quite double that for new pipes but that is perfectly consistent with what is known of fluid friction in other cases.
| Kind of Pipe. | Diameter in Metres. | Value of m. | Mean Value of m. | Authority. |
| Tin plate | 0.036 | .01697 | .01686 | Bossut |
| 0.054 | .01676 | |||
| Wrought iron | 0.016 | .01302 | .01310 | Hamilton Smith |
| 0.027 | .01319 | |||
| Asphalted pipes | 0.027 | .01749 | .01831 | Hamilton Smith |
| 0.306 | .02058 | W. W. Bonn | ||
| 0.306 | .02107 | W. W. Bonn | ||
| 0.419 | .01650 | Lampe | ||
| 1.219 | .01317 | Stearns | ||
| 1.219 | .02107 | Gale | ||
| Riveted wrought iron | 0.278 | .01370 | .01403 | Hamilton Smith |
| 0.322 | .01440 | |||
| 0.375 | .01390 | |||
| 0.432 | .01368 | |||
| 0.657 | .01448 | |||
| New cast iron | 0.082 | .01725 | .01658 | Darcy |
| 0.137 | .01427 | |||
| 0.188 | .01734 | |||
| 0.500 | .01745 | |||
| Cleaned cast iron | 0.080 | .01979 | .01994 | Darcy |
| 0.245 | .02091 | |||
| 0.297 | .01913 | |||
| Incrusted cast iron | 0.036 | .03693 | .03643 | Darcy |
| 0.080 | .03530 | |||
| 0.243 | .03706 |
- ↑ “Formulae for the Flow of Water in Pipes,” Industries (Manchester, 1886).
