§ 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.
[Illustration: FIG. 80.]
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 dl, between cross sections at z and z +
dz ft. above any horizontal datum line xx, the pressures at the cross
sections being p and p + dp lb. per square foot. Further, let Q be the
volume of flow or discharge of the pipe per second, [Omega] the area
of a normal cross section, and [chi] the perimeter of the pipe. The Q
cubic feet, which flow through the space considered per second, weigh
GQ lb., and fall through a height -dz ft. The work done by gravity is
then
-GQ dz;
a positive quantity if dz is negative, and vice versa. The resultant
pressure parallel to the axis of the pipe is p - (p + dp) = -dp lb.
per square foot of the cross section. The work of this pressure on the
volume Q is
-Q dp.
The only remaining force doing work on the system is the friction
against the surface of the pipe. The area of that surface is [chi]dl.
The work expended in overcoming the frictional resistance per second
is (see § 66, eq. 3)
-[zeta]G[chi]dlv³/2g,
or, since Q = [Omega]v,
-[zeta]G([chi]/[Omega]) Q (v²/2g) dl;
the negative sign being taken because the work is done against a
resistance. Adding all these portions of work, and equating the result
to zero, since the motion is uniform,--
-GQ dz - Q dp - [zeta]G([chi]/[Omega]) Q (v²/2g) dl = 0.
Dividing by GQ,
dz + dp/G + [zeta]([chi]/[Omega])(v²/2g) dl = 0.
Integrating,
z + p/G + [zeta]([chi]/[Omega])(v²/2g)l = constant. (1)
§ 72. Let A and B (fig. 81) be any two sections of the pipe for which
p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Then
z1 + p1/G + [zeta]([chi]/[Omega])(v²/2g)l1
= z2 + p2/G + [zeta]([chi]/[Omega])(v²/2g)l2;
or, if l2 - l1 = L, rearranging the terms,
[zeta]v²/2g = (1/L){(z1 + p1/G) - (z2 + p2/G)}[Omega]/[chi]. (2)
Suppose pressure columns introduced at A and B. The water will rise in
those columns to the heights p1/G and p2/G due to the pressures p1 and
p2 at A and B. Hence (z1 + p1/G) - (z2 + p2/G) is the quantity
represented in the figure by DE, the fall of level of the pressure
columns, or _virtual fall_ of the pipe. If there were no friction in
the pipe, then by Bernoulli's equation there would be no fall of level
of the pressure columns, the velocity being the same at A and B. Hence
DE or h is the head lost in friction in the distance AB. The quantity
DE/AB = h/L is termed the virtual slope of the pipe or virtual fall
per foot of length. It is sometimes termed very conveniently the
relative fall. It will be denoted by the symbol i.
[Illustration: FIG. 81.]
The quantity [Omega]/[chi] which appears in many hydraulic equations
is called the hydraulic mean radius of the pipe. It will be denoted by
m.
Introducing these values,
[zeta]v²/2g = mh/L = mi. (3)
For pipes of circular section, and diameter d,
m = [Omega]/[chi] = ¼[pi]d²/[pi]d = ¼d.