the p0rtion of surface which succeeds the first will be rubbing, not
against stationary water but against water partiall movin in its I
I Y E

pwn direction, and cannot therefore experience so much resistance rom it.

§ 69. The following table gives a general statement of Froude's results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated. the results obtained with the disks and Froude's results on olanks sponding to any speed N. From these the values off and n can be deduced, f being the friction per square foot at unit velocity. For comparison with Fr0ude's results it is convenient to calculate the resistance at 10 ft. per second, which is F=f10” The disks were rotated in chambers 22 in. diameter and 3, 6 and I2 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. Ilence 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 from I-9 to 2°I. This is in agreement with A V B C which the resistance was proportional varied Froude's results.

Experiments with a bright brass disk showed that the friction decreased with increase of 2 temperature. The diminution between 41° T06 4;4 4 3' and I;§ O° F. amounted to 18 % In the general 4 5 337 equation M =cN" for any given disk, I I Length of Surface, or Distance from Cutwater, in feet. W 2 ft. 8 ft. 20 ft. 50 ft.

4 A B c A B c I A B c

I Varnish 2-OO 390 325 264 278 240 I-83 -250 °226 Paraffin . 370 314 260 271 237

Tinloil 2 16 295 278 263 262 244 I-83] -246 °232 Calico I 93 I 725 626 504 531 447 I-87 — Fine sand 2 oo 690 583 450 480 3.84

Medium sand 2~OO 730 625 488 534 465 2-oo -488 -456 Coarse sand 2-00 I- 880 714 520 588 490 c¢=0-1328(I ~0-o02It),

where Q is the value of c for a bright brass Columns A give the power of the speed to which the resistance is approximately proportional.

Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table. Columns C give the resistance in pounds of a square foot of surface at the distance stern ward from the cutwater stated in the heading. Although these experiments do not directly deal with surfaces of greater length than 50 ft., they indicate what would be the resistances of longer surfaces. For at 50 ft. the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at IO ft. per second diminishes from 0~4.I to O'32 Th per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0~27S to O'25O lb per square foot for an increase from 20 ft. to 50 ft. I

If the decrease of friction stern wards is due to the generation of a current accompanying the moving plane, there is not at first sight any reason why the decrease should not be greater than that shown I by the experiments. The current accompanying the board might be assumed to gain in volume and velocity stern wards, till the velocity was nearly the same as that of the moving plane and the friction per square foot nearly zero. That this does not happen appears to be due to the mixing up of the current with the still water surrounding it. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surrounding space, and the current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the current accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also. § 70. Friction of Rotating Dis/as.-A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. Experiments carried out by Professor W. C. Unwin (Proc. Inst. Civ. Eng. lxxx.) are useful both as illustrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal umps. Disks of IO, 15 and 20 in. diameter fixed on a vertical shafi were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three Hne wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed. 1 If w is an element of area on the disk moving with the velocity v, the friction on this element is foe", where f and n are constant for any given kind of surface. Let a be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between r and I-f-df. Its area is 21rrdr, its velocity ar and the friction of this ring is f21rrdfa"r". The moment of the friction about the axis of rotation is: >1ra'{fr"'“'1dr, and the total moment of friction for the two sides of the disk is

l

l

hi = 41ro."ff§ 1"'l'2(]r = l41ra"/fn -1-3) }fR"+“. If N is the number of revolutions per sec., 1,1={2n+?7rn+1Nn/(n }j"R1|+3

and the work expended In rotating the disk is Mu. = {2"*'1r"+”N"+'/(n+3)}fR"+' foot lb per sec. The experiments give directly the values of M for the disks c0rredisk 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 50 ft. long. The values given are the resistances per square foot at I0 ft. per sec.

Froudelv Experiments. Disk Experiments. Tinfoil surface O°232 Bright brass 0~2o2 to 0~229 Varnish ., .. 0-226 Varnish 0-220 to O'233 Fine sand . 0-337 Fine sand 0-339

Medium sand 0~456 Very coarse sand 0~587 to O'7I5 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.

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 Ill, between

cross sections at

z and z-I-dz ft. above

any horizontal datum

line xx, the pressures at

the cross sections being

PJ?

p and p-I-dp lb per '?"square

foot. Further,

let Q be the volume of

$

2

-1l[ V> 5

""»-f "ff

1

"'- I

- » A /PHI

T"'

é ', ;fl'-S

524112 9

4 —.LY

FIG. 80.

flow or discharge of the pipe per second, SZ the area of a normal cross section, and X the perimeter of the pipe. The Q cubic feet, which How through the space considered per second, weigh GQ lb, and fall through a height-dz ft. The work done by gravity is then -GQdz; »

a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is P—(p-l»dp) = -dp lb per square foot of the cross section. The work of this pressure on the volume Q is

-°QfiP-

The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is X dl. The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3)

-§ GXdlv“/2g,

or, since Q==f2v, V

- § 'G(X/S?)Q(1'”, /'2§ >1U;