and the work expended in rotating the disk is
M[alpha] = {2^(n+3)[pi]^(n+2)N^(n+1)/(n + 3)} fR^(n+3), foot lb. per sec.
The experiments give directly the values of M for the disks
corresponding to any speed N. From these the values of f and n can be
deduced, f being the friction per square foot at unit velocity. For
comparison with Froude's results it is convenient to calculate the
resistance at 10 ft. per second, which is F = f10^n.
The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 in.
deep. In all cases the friction of the disks increased a little as the
chamber was made larger. This is probably due to the stilling of the
eddies against the surface of the chamber and the feeding back of the
stilled water to the disk. Hence the friction depends not only on the
surface of the disk but to some extent on the surface of the chamber
in which it rotates. If the surface of the chamber is made rougher by
covering with coarse sand there is also an increase of resistance.
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 = cN^n for any
given disk,
c_t = 0.1328(1 - 0.0021t),
where c_t 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