[zeta]v²/2g = ¼dh/L = ¼di; (4)
or
h = [zeta](4L/d)(v²/2g); (4a)
which shows that the head lost in friction is proportional to the head
due to the velocity, and is found by multiplying that head by the
coefficient 4[zeta]L/d. It is assumed above that the atmospheric
pressure at C and D is the same, and this is usually nearly the case.
But if C and D are at greatly different levels the excess of
barometric pressure at C, in feet of water, must be added to p2/G.
§ 73. _Hydraulic Gradient or Line of Virtual Slope._--Join CD. Since
the head lost in friction is proportional to L, any intermediate
pressure column between A and B will have its free surface on the line
CD, and the vertical distance between CD and the pipe at any point
measures the pressure, exclusive of atmospheric pressure, in the pipe
at that point. If the pipe were laid along the line CD instead of AB,
the water would flow at the same velocity by gravity without any
change of pressure from section to section. Hence CD is termed the
virtual slope or hydraulic gradient of the pipe. It is the line of
free surface level for each point of the pipe.
If an ordinary pipe, connecting reservoirs open to the air, rises at
any joint above the line of virtual slope, the pressure at that point
is less than the atmospheric pressure transmitted through the pipe. At
such a point there is a liability that air may be disengaged from the
water, and the flow stopped or impeded by the accumulation of air. If
the pipe rises more than 34 ft. above the line of virtual slope, the
pressure is negative. But as this is impossible, the continuity of the
flow will be broken.
If the pipe is not straight, the line of virtual slope becomes a
curved line, but since in actual pipes the vertical alterations of
level are generally small, compared with the length of the pipe,
distances measured along the pipe are sensibly proportional to
distances measured along the horizontal projection of the pipe. Hence
the line of hydraulic gradient may be taken to be a straight line
without error of practical importance.
[Illustration: FIG. 82.]
§ 74. _Case of a Uniform Pipe connecting two Reservoirs, when all the
Resistances are taken into account._--Let h (fig. 82) be the
difference of level of the reservoirs, and v the velocity, in a pipe
of length L and diameter d. The whole work done per second is
virtually the removal of Q cub. ft. of water from the surface of the
upper reservoir to the surface of the lower reservoir, that is GQh
foot-pounds. This is expended in three ways. (1) The head v²/2g,
corresponding to an expenditure of GQv²/2g foot-pounds of work, is
employed in giving energy of motion to the water. This is ultimately
wasted in eddying motions in the lower reservoir. (2) A portion of
head, which experience shows may be expressed in the form
[zeta]0v²/2g, corresponding to an expenditure of GQ[zeta]0v²/2g
foot-pounds of work, is employed in overcoming the resistance at the
entrance to the pipe. (3) As already shown the head expended in
overcoming the surface friction of the pipe is [zeta](4L/d)(v²/2g)
corresponding to GQ[zeta](4L/d)(v²/2g) foot-pounds of work. Hence
GQh = GQv²/2g + GQ[zeta]0v²/2g + GQ[zeta]·4L·v²/d·2g;
h = (1 + [zeta]0 + [zeta]·4L/d)v²/2g.
(5)
v = 8.025 [root][hd/{(1 + [zeta]0)d + 4[zeta]L}].
If the pipe is bell-mouthed, [zeta]0 is about = .08. If the entrance
to the pipe is cylindrical, [zeta]0 = 0.505. Hence 1 + [zeta]0 = 1.08
to 1.505. In general this is so small compared with [zeta]4L/d that,
for practical calculations, it may be neglected; that is, the losses
of head other than the loss in surface friction are left out of the
reckoning. It is only in short pipes and at high velocities that it is
necessary to take account of the first two terms in the bracket, as
well as the third. For instance, in pipes for the supply of turbines,
v is usually limited to 2 ft. per second, and the pipe is bellmouthed.
Then 1.08v²/2g = 0.067 ft. In pipes for towns' supply v may range from
2 to 4½ ft. per second, and then 1.5v²/2g = 0.1 to 0.5 ft. In either
case this amount of head is small compared with the whole virtual fall
in the cases which most commonly occur.
When d and v or d and h are given, the equations above are solved
quite simply. When v and h are given and d is required, it is better
to proceed by approximation. Find an approximate value of d by
assuming a probable value for [zeta] as mentioned below. Then from
that value of d find a corrected value for [zeta] and repeat the
calculation.