= GENERAL PRINCIPLES OF HYDRAULICS 463 
jount of turbulence in the water is a varying and uncertain 
itity. The consequence is that there is no exact theory of the loss of 
zy in practical cases, and the formul in use are therefore to a large 
ctent empirical. 
_ Experiment has shown that in practical cases the loss of energy is 
proximately proportional to the square of the velocity of the water and 
) the amount of the wetted surface, and inversely proportional to the 
of the cross section of the stream. The wetted surface is s/, where s 
s the wetted perimeter, hence the loss of energy is approximately pro- 
a sl 
A 
und * — 1 i, the reciprocal of the hydraulic mean radius. 
A m 
- The head or energy due to the velocity v is 
v?, where A is the area of the cross section of the stream, 
= and the loss of head 
2g 
_ may be written h’=/ - Ks . 5 where / is a coefficient to be determined by 
eriment, and is called the coefficient of friction for the pipe. This co- 
efficient f is not simply a coefficient of friction between the water and 
the surface of the pipe, but includes a coefficient of resistance due to 
eddying motions in the water itself. 
For a cireular pipe running full m=d/4, hence h’=/-—- 
If A and B (Fig. 760) are two sections of a pipe at a distance / apart, 
the heights of A and B above datum being h, and h, respectively, and 
the pressures P, and P,. If the pipe be of uniform section, then the 
velocity v will be the same throughout.» The total energy or head of the 
P 
er at A is 45 +h and the total energy or head at B is 
Ty +h, The loss of energy or head between A and B is 
29 
’ tL # ‘St a P, , 
= = — —— —_ a | =n h, h 
h’=f 3 Hence so rh wrt lg th’, 
P 
z 
and /’ = Fin *ey (hy —h,), which suggests the experimental method of 
finding h’. Knowing h’, if v is determined, f can then be found for the 
particular pipe experimented with. 
_ 405. Darcy’s Formula.—Darcy found from numerous and careful 
‘experiments on the flow of water in pipes up to 20 inches in diameter 
that the coefficient f varied with the velocity of the water, and also with 
the diameter of the pipe. Since the variation in the velocity in ordinary 
cases is comparatively small, its effect on the value of / may generally be 
“neglected, but the range of pipe diameter in practice being considerable, 
Darcy allowed for it in the formula f= 0-005(1 + - for new clean 
ipes, where d is the diameter of the pipe in feet. For old and incrusted 
pipes Darcy found the value of / to be double that for new clean pipes. 
_ 406. Hydraulic Gradient.—Referring to Fig. 760, A and B are 
two sections of a straight uniform pipe at a distance / from one another, 
* 
. 
(a 
