464 APPLIED MECHANICS ; . 
h, and h, are the heights of A and B above datum, and P, and P, re 
the pressures at A and B. ‘If pressure tubes, called plezometers, 
be inserted in the pipe at A : ; 
and B, the water will rise in these HG_____ _Horvzoniar ear 
tubes to height P,/w and P,/w, Aran} 
as shown. The line CD join- ~z 
~ 
~ 
~"< 
ing the tops of the water- 
pressure columns is called the 
hydraulic gradient or virtual 
slope of the pipe, and fh’, the JI" ---->>> 
difference between the levels of 
‘C and D, is called the virtual be 
fall of the pipe AB. Thethy: 57 SAS ae ne ba 
draulic gradient or virtual slope ic oes 
is measured by the ratio h’/l, and is denoted by 7. 
The virtual fall h’ is evidently given by the equation 
Himh, hob aoe 
> 
and is equal to the loss of head between A and B. In Art. 404 it was 
shown that h’ is given by the equation h’=/- ai Es 
m 2g 
Since the loss of head is proportional to the length of pipe, it follows | 
that for a straight uniform pipe the hydraulic gradient is a straight line, — 
When the pipe is not straight, points in the line of hydraulic gradient — 
. 
e , taking successive — 
values of 7, the length of pipe, between the points considered. D 
In water mains the curvature in the direction of the length is generally 
small, and its effect on the hydraulic gradient is generally neglected. For 
example, in Fig. 761 is shown 
a pipe AEB leading from a 
tank or reservoir at A to 
another at B at a lower level. 
The hydraulic gradient is 
shown straight, and with the 
amount of curvature shown on 
the pipe, this will be found 
to be approximately true. It 
will be noticed that the upper 
end of the line of hydraulic gradient is below the level of the water 
in the tank A, which is accounted for by the loss of head at the 
entrance to the pipe at A. 
Another point illustrated by Fig. 761 is that a part CD of the pipe 
is above the hydraulic gradient, which shows that in that part of the — 
pipe the pressure is less than atmospheric. If there is a leaky joint in~ 
CD air will rush in, and while the pipe from A to E will run full with 
a hydraulic gradient FE, the pipe from E to B will not run full, and the 
discharge will be reduced. Water pipes should therefore be arranged, if 
possible, so as to lie below the hydraulic gradient. 
When a valve or other obstruction occurs in a pipe there is a sudden 
‘4 
may be determined from the equation h’=/:- E. 
m 
Fig. 761. 
