Chapter 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 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.

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.

Then

[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.

§ 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.

The equation above may be put in the form

h = (4[zeta]/d)[{(1 + [zeta]0)d/4[zeta]} + L] v²/2g; (6)

from which it is clear that the head expended at the mouthpiece is
equivalent to that of a length

(1 + [zeta]0)d/4[zeta]

of the pipe. Putting 1 + [zeta]0 = 1.505 and [zeta] = 0.01, the length
of pipe equivalent to the mouthpiece is 37.6d nearly. This may be
added to the actual length of the pipe to allow for mouthpiece
resistance in approximate calculations.

§ 75. _Coefficient of Friction for Pipes discharging Water._--From the
average of a large number of experiments, the value of [zeta] for
ordinary iron pipes is

[zeta] = 0.007567. (7)

But practical experience shows that no single value can be taken
applicable to very different cases. The earlier hydraulicians occupied
themselves chiefly with the dependence of [zeta] on the velocity.
Having regard to the difference of the law of resistance at very low
and at ordinary velocities, they assumed that [zeta] might be
expressed in the form

[zeta] = a + [beta]/v.

The following are the best numerical values obtained for [zeta] so
expressed:--

+----------------------------------+----------+----------+
| | [alpha] | [beta] |
+----------------------------------+----------+----------+
| R. de Prony (from 51 experiments)| 0.006836 | 0.001116 |
| J. F. d'Aubuisson de Voisins | 0.00673 | 0.001211 |
| J. A. Eytelwein | 0.005493 | 0.00143 |
+----------------------------------+----------+----------+

Weisbach proposed the formula

4[zeta] = [alpha] + [beta]/[root]v = 0.003598 + 0.004289/[root]v. (8)

§ 76. _Darcy's Experiments on Friction in Pipes._--All previous
experiments on the resistance of pipes were superseded by the
remarkable researches carried out by H. P. G. Darcy (1803-1858), the
Inspector-General of the Paris water works. His experiments were
carried out on a scale, under a variation of conditions, and with a
degree of accuracy which leaves little to be desired, and the results
obtained are of very great practical importance. These results may be
stated thus:--

1. For new and clean pipes the friction varies considerably with the
nature and polish of the surface of the pipe. For clean cast iron it
is about 1½ times as great as for cast iron covered with pitch.

2. The nature of the surface has less influence when the pipes are old
and incrusted with deposits, due to the action of the water. Thus old
and incrusted pipes give twice as great a frictional resistance as new
and clean pipes. Darcy's coefficients were chiefly determined from
experiments on new pipes. He doubles these coefficients for old and
incrusted pipes, in accordance with the results of a very limited
number of experiments on pipes containing incrustations and deposits.

3. The coefficient of friction may be expressed in the form [zeta] =
[alpha] + [beta]/v; but in pipes which have been some time in use it
is sufficiently accurate to take [zeta] = [alpha]1 simply, where
[alpha]1 depends on the diameter of the pipe alone, but [alpha] and
[beta] on the other hand depend both on the diameter of the pipe and
the nature of its surface. The following are the values of the
constants.

For pipes which have been some time in use, neglecting the term
depending on the velocity;

[zeta] = [alpha](1 + [beta]/d). (9)

+-------------------------------------------------+---------+------+
| | [alpha] |[beta]|
+-------------------------------------------------+---------+------+
| For drawn wrought-iron or smooth cast-iron pipes| .004973 | .084 |
| For pipes altered by light incrustations | .00996 | .084 |
+-------------------------------------------------+---------+------+

These coefficients may be put in the following very simple form,
without sensibly altering their value:--

For clean pipes [zeta] = .005(1 + (1/12)d) (9a)
For slightly incrusted pipes [zeta] = .01(1 + (1/12)d)

_Darcy's Value of the Coefficient of Friction [zeta] for Velocities
not less than 4 in. per second._

+----------+------------------++----------+------------------+
| Diameter | [zeta] || Diameter | [zeta] |
| of Pipe +--------+---------|| of Pipe +------------------+
|in Inches.| New |Incrusted||in Inches.| New |Incrusted|
| | Pipes. | Pipes. || | Pipes. | Pipes. |
+----------+--------+---------++----------+--------+---------+
| 2 |0.00750 |0.01500 || 18 | .00528 | .01056 |
| 3 | .00667 | .01333 || 21 | .00524 | .01048 |
| 4 | .00625 | .01250 || 24 | .00521 | .01042 |
| 5 | .00600 | .01200 || 27 | .00519 | .01037 |
| 6 | .00583 | .01167 || 30 | .00517 | .01033 |
| 7 | .00571 | .01143 || 36 | .00514 | .01028 |
| 8 | .00563 | .01125 || 42 | .00512 | .01024 |
| 9 | .00556 | .01111 || 48 | .00510 | .01021 |
| 12 | .00542 | .01083 || 54 | .00509 | .01019 |
| 15 | .00533 | .01067 || | | |
+----------+--------+---------++----------+--------+---------+

These values of [zeta] are, however, not exact for widely differing
velocities. To embrace all cases Darcy proposed the expression

[zeta] = ([alpha] + [alpha]1/d) + ([beta] + [beta]1/d²)/v; (10)

which is a modification of Coulomb's, including terms expressing the
influence of the diameter and of the velocity. For clean pipes Darcy
found these values

[alpha] = .004346
[alpha]1 = .0003992
[beta] = .0010182
[beta]1 = .000005205.

It has become not uncommon to calculate the discharge of pipes by the
formula of E. Ganguillet and W. R. Kutter, which will be discussed
under the head of channels. For the value of c in the relation v = c
[root](mi), Ganguillet and Kutter take

41.6 + 1.811/n + .00281/i
c = ----------------------------------
1 + [(41.6 + .00281/i)(n/[root]m)]

where n is a coefficient depending only on the roughness of the pipe.
For pipes uncoated as ordinarily laid n = 0.013. The formula is very
cumbrous, its form is not rationally justifiable and it is not at all
clear that it gives more accurate values of the discharge than simpler
formulae.

§ 77. _Later Investigations on Flow in Pipes._--The foregoing
statement gives the theory of flow in pipes so far as it can be put in
a simple rational form. But the conditions of flow are really more
complicated than can be expressed in any rational form. Taking even
selected experiments the values of the empirical coefficient [zeta]
range from 0.16 to 0.0028 in different cases. Hence means of
discriminating the probable value of [zeta] are necessary in using the
equations for practical purposes. To a certain extent the knowledge
that [zeta] decreases with the size of the pipe and increases very
much with the roughness of its surface is a guide, and Darcy's method
of dealing with these causes of variation is very helpful. But a
further difficulty arises from the discordance of the results of
different experiments. For instance F. P. Stearns and J. M. Gale both
experimented on clean asphalted cast-iron pipes, 4 ft. in diameter.
According to one set of gaugings [zeta] = .0051, and according to the
other [zeta] = .0031. It is impossible in such cases not to suspect
some error in the observations or some difference in the condition of
the pipes not noticed by the observers.

It is not likely that any formula can be found which will give exactly
the discharge of any given pipe. For one of the chief factors in any
such formula must express the exact roughness of the pipe surface, and
there is no scientific measure of roughness. The most that can be done
is to limit the choice of the coefficient for a pipe within certain
comparatively narrow limits. The experiments on fluid friction show
that the power of the velocity to which the resistance is proportional
is not exactly the square. Also in determining the form of his
equation for [zeta] Darcy used only eight out of his seventeen series
of experiments, and there is reason to think that some of these were
exceptional. Barré de Saint-Venant was the first to propose a formula
with two constants,

dh/4l = mV^n,

where m and n are experimental constants. If this is written in the
form

log m + n log v = log (dh/4l),

we have, as Saint-Venant pointed out, the equation to a straight line,
of which m is the ordinate at the origin and n the ratio of the slope.
If a series of experimental values are plotted logarithmically the
determination of the constants is reduced to finding the straight line
which most nearly passes through the plotted points. Saint-Venant
found for n the value of 1.71. In a memoir on the influence of
temperature on the movement of water in pipes (Berlin, 1854) by G. H.
L. Hagen (1797-1884) another modification of the Saint-Venant formula
was given. This is h/l = mv^n/d^x, which involves three experimental
coefficients. Hagen found n = 1.75; x = 1.25; and m was then nearly
independent of variations of v and d. But the range of cases examined
was small. In a remarkable paper in the _Trans. Roy. Soc._, 1883,
Professor Osborne Reynolds made much clearer the change from regular
stream line motion at low velocities to the eddying motion, which
occurs in almost all the cases with which the engineer has to deal.
Partly by reasoning, partly by induction from the form of
logarithmically plotted curves of experimental results, he arrived at
the general equation h/l = c(v^n/d^(3 - n))P^(2 - n), where n = l for
low velocities and n = 1.7 to 2 for ordinary velocities. P is a
function of the temperature. Neglecting variations of temperature
Reynold's formula is identical with Hagen's if x = 3 - n. For
practical purposes Hagen's form is the more convenient.

_Values of Index of Velocity._

+--------------------+---------------+----------+---------------+
| | | Diameter | |
| Surface of Pipe. | Authority. | of Pipe | Values of n. |
| | |in Metres.| |
+--------------------+---------------+----------+---------------+
| Tin plate | Bossut | /.036 | 1.697 \ 1.72 |
| | | \.054 | 1.730 / |
| | | | |
| Wrought iron (gas | Hamilton Smith| /.0159 | 1.756 \ 1.75 |
| pipe) | | \.0267 | 1.770 / |
| | | | |
| | | /.014 | 1.866 \ |
| Lead | Darcy | < .027 | 1.755 > 1.77 |
| | | \.041 | 1.760 / |
| | | | |
| Clean brass | Mair | .036 | 1.795 1.795|
| | | | |
| / | Hamilton Smith| / .0266 | 1.760 \ |
| Asphalted < | Lampe. |< .4185 | 1.850 > 1.85 |
| | | W. W. Bonn | | .306 | 1.582 | |
| \ | Stearns | \1.219 | 1.880 / |
| | | | |
| Riveted wrought \ | | /.2776 | 1.804 \ |
| iron > | Hamilton Smith|< .3219 | 1.892 > 1.87 |
| / | | \.3749 | 1.852 / |
| | | | |
| Wrought iron (gas\ | | /.0122 | 1.900 \ |
| pipe) >| Darcy |< .0266 | 1.899 > 1.87 |
| / | | \.0395 | 1.838 / |
| | | | |
| | | /.0819 | 1.950 \ |
| New cast iron | Darcy |< .137 | 1.923 > 1.95 |
| | | |.188 | 1.957 | |
| | | \.50 | 1.950 / |
| | | | |
| | | /.0364 | 1.835 \ |
| | | |.0801 | 2.000 > 2.00 |
| Cleaned cast iron | Darcy |< .2447 | 2.000 | |
| | | \.397 | 2.07 / |
| | | | |
| | | /.0359 | 1.980 \ |
| Incrusted cast iron| Darcy |< .0795 | 1.990 > 2.00 |
| | | \.2432 | 1.990 / |
+--------------------+---------------+----------+---------------+

In 1886, Professor W. C. Unwin plotted logarithmically all the most
trustworthy experiments on flow in pipes then available.[5] Fig. 83
gives one such plotting. The results of measuring the slopes of the
lines drawn through the plotted points are given in the table.

It will be seen that the values of the index n range from 1.72 for the
smoothest and cleanest surface, to 2.00 for the roughest. The numbers
after the brackets are rounded off numbers.

The value of n having been thus determined, values of m/d^x were next
found and averaged for each pipe. These were again plotted
logarithmically in order to find a value for x. The lines were not
very regular, but in all cases the slope was greater than 1 to 1, so
that the value of x must be greater than unity. The following table
gives the results and a comparison of the value of x and Reynolds's
value 3 - n.

+-----------------------+--------+--------+-------+
| Kind of Pipe. | n | 3 - n | x |
+-----------------------+--------+--------+-------+
| Tin plate | 1.72 | 1.28 | 1.100 |
| Wrought iron (Smith) | 1.75 | 1.25 | 1.210 |
| Asphalted pipes | 1.85 | 1.15 | 1.127 |
| Wrought iron (Darcy) | 1.87 | 1.13 | 1.680 |
| Riveted wrought iron | 1.87 | 1.13 | 1.390 |
| New cast iron | 1.95 | 1.05 | 1.168 |
| Cleaned cast iron | 2.00 | 1.00 | 1.168 |
| Incrusted cast iron | 2.00 | 1.00 | 1.160 |
+-----------------------+--------+--------+-------+

With the exception of the anomalous values for Darcy's wrought-iron
pipes, there is no great discrepancy between the values of x and 3 -
n, but there is no appearance of relation in the two quantities. For
the present it appears preferable to assume that x is independent of
n.

It is now possible to obtain values of the third constant m, using the
values found for n and x. The following table gives the results, the
values of m being for metric measures.

Here, considering the great range of diameters and velocities in the
experiments, the constancy of m is very satisfactorily close. The
asphalted pipes give rather variable values. But, as some of these
were new and some old, the variation is, perhaps, not surprising. The
incrusted pipes give a value of m quite double that for new pipes but
that is perfectly consistent with what is known of fluid friction in
other cases.

+---------------+----------+-----------+----------+----------------+
| | Diameter | Value of | Mean | |
| Kind of Pipe. | in | m. | Value | Authority. |
| | Metres. | | of m. | |
+---------------+----------+-----------+----------+----------------+
| Tin plate | / 0.036 | .01697 \ | .01686 | Bossut |
| | \ 0.054 | .01676 / | | |
| | | | | |
| Wrought iron | / 0.016 | .01302 \ | .01310 | Hamilton Smith |
| | \ 0.027 | .01319 / | | |
| | | | | |
| | / 0.027 | .01749 \ | / | Hamilton Smith |
| | | 0.306 | .02058 | | | | W. W. Bonn |
| Asphalted | < 0.306 | .02107 > | .01831< | W. W. Bonn |
| pipes | | 0.419 | .01650 | | | | Lampe |
| | | 1.219 | .01317 | | | | Stearns |
| | \ 1.219 | .02107 / | \ | Gale |
| | | | | |
| | / 0.278 | .01370 \ | | |
| | | 0.322 | .01440 | | | |
| Riveted | < 0.375 | .01390 > | .01403 | Hamilton Smith |
| wrought iron| | 0.432 | .01368 | | | |
| | \ 0.657 | .01448 / | | |
| | | | | |
| | / 0.082 | .01725 \ | | |
| New cast iron | < 0.137 | .01427 > | .01658 | Darcy |
| | | 0.188 | .01734 | | | |
| | \ 0.500 | .01745 / | | |
| | | | | |
| Cleaned cast | / 0.080 | .01979 \ | | |
| iron | < 0.245 | .02091 > | .01994 | Darcy |
| | \ 0.297 | .01913 / | | |
| | | | | |
| Incrusted cast| / 0.036 | .03693 \ | | |
| iron | < 0.080 | .03530 > | .03643 | Darcy |
| | \ 0.243 | .03706 / | | |
+---------------+----------+-----------+----------+----------------+

_General Mean Values of Constants._

The general formula (Hagen's)--h/l = mv^n/d^x.2g--can therefore be
taken to fit the results with convenient closeness, if the following
mean values of the coefficients are taken, the unit being a metre:--

+----------------------+-------+-------+------+
| Kind of Pipe. | m | x | n |
+----------------------+-------+-------+------+
| Tin plate | .0169 | 1.10 | 1.72 |
| Wrought iron | .0131 | 1.21 | 1.75 |
| Asphalted iron | .0183 | 1.127 | 1.85 |
| Riveted wrought iron | .0140 | 1.390 | 1.87 |
| New cast iron | .0166 | 1.168 | 1.95 |
| Cleaned cast iron | .0199 | 1.168 | 2.0 |
| Incrusted cast iron | .0364 | 1.160 | 2.0 |
+----------------------+-------+-------+------+

The variation of each of these coefficients is within a comparatively
narrow range, and the selection of the proper coefficient for any
given case presents no difficulty, if the character of the surface of
the pipe is known.

It only remains to give the values of these coefficients when the
quantities are expressed in English feet. For English measures the
following are the values of the coefficients:--

+----------------------+-------+-------+------+
| Kind of Pipe. | m | x | n |
+----------------------+-------+-------+------+
| Tin plate | .0265 | 1.10 | 1.72 |
| Wrought iron | .0226 | 1.21 | 1.75 |
| Asphalted iron | .0254 | 1.127 | 1.85 |
| Riveted wrought iron | .0260 | 1.390 | 1.87 |
| New cast iron | .0215 | 1.168 | 1.95 |
| Cleaned cast iron | .0243 | 1.168 | 2.0 |
| Incrusted cast iron | .0440 | 1.160 | 2.0 |
+----------------------+-------+-------+------+

§ 78. _Distribution of Velocity in the Cross Section of a
Pipe._--Darcy made experiments with a Pitot tube in 1850 on the
velocity at different points in the cross section of a pipe. He
deduced the relation

V - v = 11.3(r^(3/2)/R) [root]i,

where V is the velocity at the centre and v the velocity at radius r
in a pipe of radius R with a hydraulic gradient i. Later Bazin
repeated the experiments and extended them (_Mém. de l'Académie des
Sciences_, xxxii. No. 6). The most important result was the ratio of
mean to central velocity. Let b = Ri/U², where U is the mean velocity
in the pipe; then V/U = 1 + 9.03 [root]b. A very useful result for
practical purposes is that at 0.74 of the radius of the pipe the
velocity is equal to the mean velocity. Fig. 84 gives the velocities
at different radii as determined by Bazin.

§ 79. _Influence of Temperature on the Flow through Pipes._--Very
careful experiments on the flow through a pipe 0.1236 ft. in diameter
and 25 ft. long, with water at different temperatures, have been made
by J. G. Mair (_Proc. Inst. Civ. Eng._ lxxxiv.). The loss of head was
measured from a point 1 ft. from the inlet, so that the loss at entry
was eliminated. The 1½ in. pipe was made smooth inside and to gauge,
by drawing a mandril through it. Plotting the results logarithmically,
it was found that the resistance for all temperatures varied very
exactly as v^(1.795), the index being less than 2 as in other
experiments with very smooth surfaces. Taking the ordinary equation of
flow h = [zeta](4L/D)(v²/2g), then for heads varying from 1 ft. to
nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft.
per second, the values of [zeta] were as follows:--

Temp. F. [zeta] | Temp. F. [zeta]
57 .0044 to .0052 | 100 .0039 to .0042
70 .0042 to .0045 | 110 .0037 to .0041
80 .0041 to .0045 | 120 .0037 to .0041
90 .0040 to .0045 | 130 .0035 to .0039
| 160 .0035 to .0038

This shows a marked decrease of resistance as the temperature rises.
If Professor Osborne Reynolds's equation is assumed h = mLV^n/d^(3 -
n), and n is taken 1.795, then values of m at each temperature are
practically constant--

Temp. F. m. | Temp. F. m.
57 0.000276 | 100 0.000244
70 0.000263 | 110 0.000235
80 0.000257 | 120 0.000229
90 0.000250 | 130 0.000225
| 160 0.000206

where again a regular decrease of the coefficient occurs as the
temperature rises. In experiments on the friction of disks at
different temperatures Professor W. C. Unwin found that the resistance
was proportional to constant × (1 - 0.0021t) and the values of m given
above are expressed almost exactly by the relation

m = 0.000311(1 - 0.00215 t).

In tank experiments on ship models for small ordinary variations of
temperature, it is usual to allow a decrease of 3% of resistance for
10° F. increase of temperature.

§ 80. _Influence of Deposits in Pipes on the Discharge. Scraping Water
Mains._--The influence of the condition of the surface of a pipe on
the friction is shown by various facts known to the engineers of
waterworks. In pipes which convey certain kinds of water, oxidation
proceeds rapidly and the discharge is considerably diminished. A main
laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in.
and 8-in. pipes. It was not protected from corrosion by any coating.
But it was found to the surprise of the engineer that in eight years
the discharge had diminished to 51% of the original discharge. J. G.
Appold suggested an apparatus for scraping the interior of the pipe,
and this was constructed and used under the direction of William
Froude (see "Incrustation of Iron Pipes," by W. Ingham, _Proc. Inst.
Mech. Eng._, 1899). It was found that by scraping the interior of the
pipe the discharge was increased 56%. The scraping requires to be
repeated at intervals. After each scraping the discharge diminishes
rather rapidly to 10% and afterwards more slowly, the diminution in a
year being about 25%.

Fig. 85 shows a scraper for water mains, similar to Appold's but
modified in details, as constructed by the Glenfield Company, at
Kilmarnock. A is a longitudinal section of the pipe, showing the
scraper in place; B is an end view of the plungers, and C, D sections
of the boxes placed at intervals on the main for introducing or
withdrawing the scraper. The apparatus consists of two plungers,
packed with leather so as to fit the main pretty closely. On the
spindle of these plungers are fixed eight steel scraping blades, with
curved scraping edges fitting the surface of the main. The apparatus
is placed in the main by removing the cover from one of the boxes
shown at C, D. The cover is then replaced, water pressure is admitted
behind the plungers, and the apparatus driven through the main. At
Lancaster after twice scraping the discharge was increased 56½%, at
Oswestry 54½%. The increased discharge is due to the diminution of the
friction of the pipe by removing the roughnesses due to oxidation. The
scraper can be easily followed when the mains are about 3 ft. deep by
the noise it makes. The average speed of the scraper at Torquay is
2(1/3) m. per hour. At Torquay 49% of the deposit is iron rust, the
rest being silica, lime and organic matter.

In the opinion of some engineers it is inadvisable to use the scraper.
The incrustation is only temporarily removed, and if the use of the
scraper is continued the life of the pipe is reduced. The only
treatment effective in preventing or retarding the incrustation due to
corrosion is to coat the pipes when hot with a smooth and perfect
layer of pitch. With certain waters such as those derived from the
chalk the incrustation is of a different character, consisting of
nearly pure calcium carbonate. A deposit of another character which
has led to trouble in some mains is a black slime containing a good
deal of iron not derived from the pipes. It appears to be an organic
growth. Filtration of the water appears to prevent the growth of the
slime, and its temporary removal may be effected by a kind of brush
scraper devised by G. F. Deacon (see "Deposits in Pipes," by Professor
J. C. Campbell Brown, _Proc. Inst. Civ. Eng._, 1903-1904).

§ 81. _Flow of Water through Fire Hose._--The hose pipes used for fire
purposes are of very varied character, and the roughness of the
surface varies. Very careful experiments have been made by J. R.
Freeman (_Am. Soc. Civ. Eng._ xxi., 1889). It was noted that under
pressure the diameter of the hose increased sufficiently to have a
marked influence on the discharge. In reducing the results the true
diameter has been taken. Let v = mean velocity in ft. per sec.; r =
hydraulic mean radius or one-fourth the diameter in feet; i =
hydraulic gradient. Then v = n[root](ri).

+---------------+---------+---------+-------+-------+-------+
| | Diameter| Gallons | | | |
| | in | (United | | | |
| | Inches. | States) | i | v | n |
| | | per min.| | | |
+---------------+---------+---------+-------+-------+-------+
| Solid rubber | 2.65 | 215 | .1863 | 12.50 | 123.3 |
| hose | " | 344 | .4714 | 20.00 | 124.0 |
| | | | | | |
| Woven cotton, | 2.47 | 200 | .2464 | 13.40 | 119.1 |
| rubber lined | " | 299 | .5269 | 20.00 | 121.5 |
| | | | | | |
| Woven cotton, | 2.49 | 200 | .2427 | 13.20 | 117.7 |
| rubber lined | " | 319 | .5708 | 21.00 | 122.1 |
| | | | | | |
| Knit cotton, | 2.68 | 132 | .0809 | 7.50 | 111.6 |
| rubber lined | " | 299 | .3931 | 17.00 | 114.8 |
| | | | | | |
| Knit cotton, | 2.69 | 204 | .2357 | 11.50 | 100.1 |
| rubber lined | " | 319 | .5165 | 18.00 | 105.8 |
| | | | | | |
| Woven cotton, | 2.12 | 154 | .3448 | 14.00 | 113.4 |
| rubber lined | " | 240 | .7673 | 21.81 | 118.4 |
| | | | | | |
| Woven cotton, | 2.53 | 54.8 | .0261 | 3.50 | 94.3 |
| rubber lined | " | 298 | .8264 | 19.00 | 91.0 |
| | | | | | |
| Unlined linen | 2.60 | 57.9 | .0414 | 3.50 | 73.9 |
| hose | " | 331 |1.1624 | 20.00 | 79.6 |
+---------------+---------+---------+-------+-------+-------+

§ 82. _Reduction of a Long Pipe of Varying Diameter to an Equivalent
Pipe of Uniform Diameter. Dupuit's Equation._--Water mains for the
supply of towns often consist of a series of lengths, the diameter
being the same for each length, but differing from length to length.
In approximate calculations of the head lost in such mains, it is
generally accurate enough to neglect the smaller losses of head and to
have regard to the pipe friction only, and then the calculations may
be facilitated by reducing the main to a main of uniform diameter, in
which there would be the same loss of head. Such a uniform main will
be termed an equivalent main.

In fig. 86 let A be the main of variable diameter, and B the
equivalent uniform main. In the given main of variable diameter A, let

l1, l2... be the lengths,
d1, d2... the diameters,
v1, v2... the velocities,
i1, i2... the slopes,

for the successive portions, and let l, d, v and i be corresponding
quantities for the equivalent uniform main B. The total loss of head
in A due to friction is

h = i1l1 + i2l2 + ...
= [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...

and in the uniform main

il = [zeta](v²·4l/2gd).

If the mains are equivalent, as defined above,

[zeta](v²·4l/2gd) = [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...

But, since the discharge is the same for all portions,

¼[pi]d²v = ¼[pi]d1²v1 = ¼[pi]d2²v2 = ...

v1 = vd²/d1²; v2 = vd²/d2² ...

Also suppose that [zeta] may be treated as constant for all the pipes.
Then

l/d = (d^4/d1^4)(l1/d1) + (d^4/d2^4(12/d2) + ...

l = (d^5/d1^5)l1 + (d^5/d2^5)l2 + ...

which gives the length of the equivalent uniform main which would have
the same total loss of head for any given discharge.

§ 83. _Other Losses of Head in Pipes._--Most of the losses of head in
pipes, other than that due to surface friction against the pipe, are
due to abrupt changes in the velocity of the stream producing eddies.
The kinetic energy of these is deducted from the general energy of
translation, and practically wasted.

_Sudden Enlargement of Section._--Suppose a pipe enlarges in section
from an area [omega]0 to an area [omega]1 (fig. 87); then

v1/v0 = [omega]0/[omega]1;

or, if the section is circular,

v1/v0 = (d0/d1)².

The head lost at the abrupt change of velocity has already been shown
to be the head due to the relative velocity of the two parts of the
stream. Hence head lost

[h]_e = (v0 - v1)²/2g = ([omega]1/[omega]0 - 1)²v1²/2g
= {(d1/d0)² - 1}² v1²/2g

or

[h]_e = [zeta]_ev1²/2g, (1)

if [zeta]_e is put for the expression in brackets.

+--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+
| [omega]1/ |1.1 |1.2 |1.5 |1.7 |1.8 |1.9 |2.0 |2.5 |3.0 |3.5 |4.0 | 5.0 | 6.0 | 7.0 | 8.0 |
| [omega]0 = | | | | | | | | | | | | | | | |
| d1/d0 = |1.05|1.10|1.22|1.30|1.34|1.38|1.41|1.58|1.73|1.87|2.00| 2.24| 2.45| 2.65| 2.83|
| | | | | | | | | | | | | | | | |
| [zeta]_e = | .01| .04| .25| .49| .64| .81|1.00|2.25|4.00|6.25|9.00|16.00|25.00|36.0 |49.0 |
+--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+

_Abrupt Contraction of Section._--When water passes from a larger to a
smaller section, as in figs. 88, 89, a contraction is formed, and the
contracted stream abruptly expands to fill the section of the pipe.
Let [omega] be the section and v the velocity of the stream at bb. At
aa the section will be c_c[omega], and the velocity
([omega]/c_c[omega])v = v/c1, where c_c is the coefficient of
contraction. Then the head lost is

[h]_m = (v/c_c - v)²/2g = (1/c_c - 1)²v²/2g;

and, if c_c is taken 0.64,

[h]_m = 0.316 v²/2g. (2)

The value of the coefficient of contraction for this case is, however,
not well ascertained, and the result is somewhat modified by friction.
For water entering a cylindrical, not bell-mouthed, pipe from a
reservoir of indefinitely large size, experiment gives

[h]_a = 0.505 v²/2g. (3)

If there is a diaphragm at the mouth of the pipe as in fig. 89, let
[omega]1 be the area of this orifice. Then the area of the contracted
stream is c_c[omega]1, and the head lost is

[h]_c = {([omega]/c_c[omega]1) - 1}²v²/2g
= [zeta]_cv²/2g (4)

if [zeta], is put for {([omega]/c_c[omega]1) - 1}². Weisbach has found
experimentally the following values of the coefficient, when the
stream approaching the orifice was considerably larger than the
orifice:--

+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
| [omega]1/[omega] = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 |0.7 | 0.8 | 0.9 | 1.0 |
| | | | | | | | | | | |
| c_c = | .616 | .614 | .612 |.610 |.617 |.605 |.603 |.601 |.598 |.596 |
| | | | | | | | | | | |
| [zeta]_c = | 231.7 |50.99 |19.78 |9.612|5.256|3.077|1.876|1.169|0.734|0.480|
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+

When a diaphragm was placed in a tube of uniform section (fig. 90) the
following values were obtained, [omega]1 being the area of the orifice
and [omega] that of the pipe:--

+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
| [omega]1/[omega] = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
| | | | | | | | | | | |
| c_c = | .624 | .632 | .643 |.659 |.681 |.712 |.755 |.813 |.892 |1.00 |
| | | | | | | | | | | |
| [xi]_c = | 225.9 |47.77 |30.83 |7.801|1.753|1.796|.797 |.290 |.060 |.000 |
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+

Elbows.--Weisbach considers the loss of head at elbows (fig. 91) to be
due to a contraction formed by the stream. From experiments with a
pipe 1¼ in. diameter, he found the loss of head

[h]_e = [zeta]_e v²/2g; (5)

[zeta]_e = 0.9457 sin² ½[phi] + 2.047 sin^4 ½[phi].

+------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| [phi] = | 20° | 40° | 60° | 80° | 90° | 100°| 110°| 120°| 130°| 140°|
| | | | | | | | | | | |
| [zeta]_e = |0.046|0.139|0.364|0.740|0.984|1.260|1.556|1.861|2.158|2.431|
+------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

Hence at a right-angled elbow the whole head due to the velocity very
nearly is lost.

_Bends._--Weisbach traces the loss of head at curved bends to a
similar cause to that at elbows, but the coefficients for bends are
not very satisfactorily ascertained. Weisbach obtained for the loss of
head at a bend in a pipe of circular section

[h]_b = [zeta]_b v²/2g; (6)

[zeta]_b = 0.131 + 1.847(d/2[rho])^(7/2),

where d is the diameter of the pipe and [rho] the radius of curvature
of the bend. The resistance at bends is small and at present very ill
determined.

_Valves, Cocks and Sluices._--These produce a contraction of the
water-stream, similar to that for an abrupt diminution of section
already discussed. The loss of head may be taken as before to be

[h]_v = [zeta]_v v²/2g; (7)

where v is the velocity in the pipe beyond the valve and [zeta]_v a
coefficient determined by experiment. The following are Weisbach's
results.

_Sluice in Pipe of Rectangular Section_ (fig. 92). Section at sluice =
[omega]1 in pipe = [omega].

+--------------------+----+----+----+----+----+----+----+----+----+-----+
| [omega]1/[omega] = |1.0 |0.9 |0.8 |0.7 |0.6 |0.5 |0.4 | 0.3| 0.2| 0.1 |
| | | | | | | | | | | |
| [zeta]_v = |0.00|.09 |.39 |.95 |2.08|4.02|8.12|17.8|44.5| 193 |
+--------------------+----+----+----+----+----+----+----+----+----+-----+

_Sluice in Cylindrical Pipe_ (fig. 93).

+-----------------------+----+-----+-----+-----+-----+-----+------+------+
| Ratio of height of | | | | | | | | |
| opening to diameter | 1.0| 7/8 | 3/4 | 5/8 | ½ | 3/8 | ¼ | 1/5 |
| of pipe | | | | | | | | |
| [omega]1/[omega] = |1.00|0.948|.856 |.740 |.609 |.466 | .315 | .159 |
| | | | | | | | | |
| [zeta]_v = |0.00|0.07 |0.26 |0.81 |2.06 |5.52 | 17.0 | 97.8 |
+-----------------------+----+-----+-----+-----+-----+-----+------+------+

_Cock in a Cylindrical Pipe_ (fig. 94). Angle through which cock is
turned = [theta].

+------------+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 5° | 10° | 15° | 20° | 25° | 30° | 35° |
| Ratio of | | | | | | | |
| cross |.926 |.850 |.772 |.692 |.613 |.535 |.458 |
| sections | | | | | | | |
| [zeta]_v = | .05 | .29 | .75 |1.56 |3.10 |5.47 |9.68 |
+------------+-----+-----+-----+-----+-----+-----+-----+

+------------+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 40° | 45° | 50° | 55° | 60° | 65° | 82° |
| Ratio of | | | | | | | |
| cross |.385 |.315 |.250 |.190 |.137 |.091 | 0 |
| sections | | | | | | | |
| [zeta]_v = | 17.3| 31.2| 52.6|106 |206 |486 |[oo] |
+------------+-----+-----+-----+-----+-----+-----+-----+

_Throttle Valve in a Cylindrical Pip_e (fig. 95)

+------------+-----+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 5° | 10° | 15° | 20° | 25° | 30° | 35° | 40° |
| | | | | | | | | |
| [zeta]_v = | .24 | .52 | .90 | 1.54| 2.51| 3.91| 6.22| 10.8|
+------------+-----+-----+-----+-----+-----+-----+-----+-----+

+------------+------+------+------+------+------+------+------+
| [theta] = | 45° | 50° | 55° | 60° | 65° | 70° | 90° |
| | | | | | | | |
| [zeta]_v = | 18.7 | 32.6 | 58.8 | 118 | 256 | 751 | [oo] |
+------------+------+------+------+------+------+------+------+

§ 84. _Practical Calculations on the Flow of Water in Pipes._--In the
following explanations it will be assumed that the pipe is of so great
a length that only the loss of head in friction against the surface of
the pipe needs to be considered. In general it is one of the four
quantities d, i, v or Q which requires to be determined. For since the
loss of head h is given by the relation h = il, this need not be
separately considered.

There are then three equations (see eq. 4, § 72, and 9a, § 76) for the
solution of such problems as arise:--

[zeta] = [alpha](1 + 1/12d); (1)

where [alpha] = 0.005 for new and = 0.01 for incrusted pipes.

[zeta]v²/2g = ¼di. (2)

Q = ¼[pi]d²v. (3)

_Problem 1._ Given the diameter of the pipe and its virtual slope, to
find the discharge and velocity of flow. Here d and i are given, and Q
and v are required. Find [zeta] from (1); then v from (2); lastly Q
from (3). This case presents no difficulty.

By combining equations (1) and (2), v is obtained directly:--

v = [root](gdi/2[zeta]) = [root](g/2[alpha]) [root][di/{1 + 1/12d}]. (4)

For new pipes [root](g/2[alpha]) = 56.72
For incrusted pipes = 40.13

For pipes not less than 1, or more than 4 ft. in diameter, the mean
values of [zeta] are

For new pipes 0.00526
For incrusted pipes 0.01052.

Using these values we get the very simple expressions--

v = 55.31 [root](di) for new pipes
= 39.11 [root](di) for incrusted pipes. (4a)

Within the limits stated, these are accurate enough for practical
purposes, especially as the precise value of the coefficient [zeta]
cannot be known for each special case.

_Problem 2._ Given the diameter of a pipe and the velocity of flow, to
find the virtual slope and discharge. The discharge is given by (3);
the proper value of [zeta] by (1); and the virtual slope by (2). This
also presents no special difficulty.

_Problem 3._ Given the diameter of the pipe and the discharge, to find
the virtual slope and velocity. Find v from (3); [zeta] from (1);
lastly i from (2). If we combine (1) and (2) we get

i = [zeta](v²/2g) (4/d) = 2a{1 + 1/12d} v²/gd; (5)

and, taking the mean values of [zeta] for pipes from 1 to 4 ft.
diameter, given above, the approximate formulae are

i = 0.0003268 v²/d for new pipes
= 0.0006536 v²/d for incrusted pipes. (5a)

_Problem 4._ Given the virtual slope and the velocity, to find the
diameter of the pipe and the discharge. The diameter is obtained from
equations (2) and (1), which give the quadratic expression

d² - d(2[alpha]v²/gi) - [alpha]v²/6gi = 0.

.: d = [alpha]v²/gi + [root]{([alpha]v²/gi) ([alpha]v²/gi + 1/6)}. (6)

For practical purposes, the approximate equations

d = 2[alpha]v²/gi + 1/12 (6a)
= 0.00031 v²/i + .083 for new pipes
= 0.00062 v²/i + .083 for incrusted pipes

are sufficiently accurate.

_Problem 5._ Given the virtual slope and the discharge, to find the
diameter of the pipe and velocity of flow. This case, which often
occurs in designing, is the one which is least easy of direct
solution. From equations (2) and (3) we get--

d^5 = 32[zeta]Q²/g[pi]²i. (7)

If now the value of [zeta] in (1) is introduced, the equation becomes
very cumbrous. Various approximate methods of meeting the difficulty
may be used.

(a) Taking the mean values of [zeta] given above for pipes of 1 to 4
ft. diameter we get

d = [root 5](32[zeta]/g[pi]²) [root 5](Q²/i) (8)
= 0.2216 [root 5](Q²/i) for new pipes
= 0.2541 [root 5](Q²/i) for incrusted pipes;

equations which are interesting as showing that when the value of
[zeta] is doubled the diameter of pipe for a given discharge is only
increased by 13%.

(b) A second method is to obtain a rough value of d by assuming [zeta]
= [alpha]. This value is

d´ = [root 5](32Q²/g[pi]²i) [root 5][alpha]
= 0.6319 [root 5](Q²/i) [root 5][alpha].

Then a very approximate value of [zeta] is

[zeta]´ = [alpha](1 + 1/12d´);

and a revised value of d, not sensibly differing from the exact value,
is

d´´ = [root 5](32Q²/g[pi]²i) [root 5][zeta]´
= 0.6319 [root 5](Q²/i) [root 5][zeta]´.

(c) Equation 7 may be put in the form

d = [root 5](32[alpha]Q²/g[pi]²i) [root 5](1 + 1/12d). (9)

Expanding the term in brackets,

[root 5](1 + 1/12d) = 1 + 1/60d - 1/1800d² ...

Neglecting the terms after the second,

d = [root 5](32[alpha]/g[pi]²) [root 5](Q²/i)·{1 + 1/60d}
= [root 5](32a/g[pi]²) [root 5](Q²/i) + 0.01667; (9a)

and

[root 5](32a/g[pi]²) = 0.219 for new pipes
= 0.252 for incrusted pipes.

§ 85. _Arrangement of Water Mains for Towns' Supply._--Town mains are
usually supplied oy gravitation from a service reservoir, which in
turn is supplied by gravitation from a storage reservoir or by pumping
from a lower level. The service reservoir should contain three days'
supply or in important cases much more. Its elevation should be such
that water is delivered at a pressure of at least about 100 ft. to the
highest parts of the district. The greatest pressure in the mains is
usually about 200 ft., the pressure for which ordinary pipes and
fittings are designed. Hence if the district supplied has great
variations of level it must be divided into zones of higher and lower
pressure. Fig. 96 shows a district of two zones each with its service
reservoir and a range of pressure in the lower district from 100 to
200 ft. The total supply required is in England about 25 gallons per
head per day. But in many towns, and especially in America, the supply
is considerably greater, but also in many cases a good deal of the
supply is lost by leakage of the mains. The supply through the branch
mains of a distributing system is calculated from the population
supplied. But in determining the capacity of the mains the fluctuation
of the demand must be allowed for. It is usual to take the maximum
demand at twice the average demand. Hence if the average demand is 25
gallons per head per day, the mains should be calculated for 50
gallons per head per day.

§ 86. _Determination of the Diameters of Different Parts of a Water
Main._--When the plan of the arrangement of mains is determined upon,
and the supply to each locality and the pressure required is
ascertained, it remains to determine the diameters of the pipes. Let
fig. 97 show an elevation of a main ABCD ..., R being the reservoir
from which the supply is derived. Let NN be the datum line of the
levelling operations, and H_a, H_b ... the heights of the main above
the datum line, H_r being the height of the water surface in the
reservoir from the same datum. Set up next heights AA1, BB1, ...
representing the minimum pressure height necessary for the adequate
supply of each locality. Then A1B1C1D1 ... is a line which should form
a lower limit to the line of virtual slope. Then if heights [h]_a,
[h]_b, [h]_c ... are taken representing the actual losses of head in
each length l_a, l_b, l_c ... of the main, A0B0C0 will be the line of
virtual slope, and it will be obvious at what points such as D0 and
E0, the pressure is deficient, and a different choice of diameter of
main is required. For any point z in the length of the main, we have

Pressure height = H_r - H_z - ([h]_a + [h]_b + ... [h]_z).

Where no other circumstance limits the loss of head to be assigned to
a given length of main, a consideration of the safety of the main from
fracture by hydraulic shock leads to a limitation of the velocity of
flow. Generally the velocity in water mains lies between 1½ and 4½ ft.
per second. Occasionally the velocity in pipes reaches 10 ft. per
second, and in hydraulic machinery working under enormous pressures
even 20 ft. per second. Usually the velocity diminishes along the main
as the discharge diminishes, so as to reduce somewhat the total loss
of head which is liable to render the pressure insufficient at the end
of the main.

J. T. Fanning gives the following velocities as suitable in pipes for
towns' supply:--

Diameter in inches 4 8 12 18 24 30 36
Velocity in feet per sec. 2.5 3.0 3.5 4.5 5.3 6.2 7.0

§ 87. _Branched Pipe connecting Reservoirs at Different Levels._--Let
A, B, C (fig. 98) be three reservoirs connected by the arrangement of
pipes shown,--l1, d1, Q1, v1; l2, d2, Q2, v2; h3, d3, Q3, v3 being the
length, diameter, discharge and velocity in the three portions of the
main pipe. Suppose the dimensions and positions of the pipes known and
the discharges required.

If a pressure column is introduced at X, the water will rise to a
height XR, measuring the pressure at X, and aR, Rb, Rc will be the
lines of virtual slope. If the free surface level at R is above b, the
reservoir A supplies B and C, and if R is below b, A and B supply C.
Consequently there are three cases:--

I. R above b; Q1 = Q2 + Q3.
II. R level with b; Q1 = Q3; Q2 = 0
III. R below b; Q1 + Q2 = Q3.

To determine which case has to be dealt with in the given conditions,
suppose the pipe from X to B closed by a sluice. Then there is a
simple main, and the height of free surface h´ at X can be determined.
For this condition

h_a - h´ = [zeta](v1²/2g)(4l1/d1)
= 32[zeta]Q´² l1/g[pi]²d1^5;

h´ - h_c = [zeta](v3²/2g)(4l3/d3)
= 32[zeta]Q´²l3/g[pi]²d3^5;

where Q´ is the common discharge of the two portions of the pipe.
Hence

(h_a - h´)/(h´ - h_c) = l1d3^5/l3d1^5,

from which h´ is easily obtained. If then h´ is greater than hb,
opening the sluice between X and B will allow flow towards B, and the
case in hand is case I. If h´ is less than h_b, opening the sluice
will allow flow from B, and the case is case III. If h´ = h_b, the
case is case II., and is already completely solved.

The true value of h must lie between h´ and h_b. Choose a new value of
h, and recalculate Q1, Q2, Q3. Then if

Q1 > Q2 + Q3 in case I.,

or

Q1 + Q2 > Q3 in case III.,

the value chosen for h is too small, and a new value must be chosen.

If

Q1 < Q2 + Q3 in case I.,

or

Q1 + Q2 < Q3 in case III.,

the value of h is too great.

Since the limits between which h can vary are in practical cases not
very distant, it is easy to approximate to values sufficiently
accurate.

§ 88. _Water Hammer._--If in a pipe through which water is flowing a
sluice is suddenly closed so as to arrest the forward movement of the
water, there is a rise of pressure which in some cases is serious
enough to burst the pipe. This action is termed water hammer or water
ram. The fluctuation of pressure is an oscillating one and gradually
dies out. Care is usually taken that sluices should only be closed
gradually and then the effect is inappreciable. Very careful
experiments on water hammer were made by N. J. Joukowsky at Moscow in
1898 (_Stoss in Wasserleitungen_, St Petersburg, 1900), and the
results are generally confirmed by experiments made by E. B. Weston
and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in.
diameter, from 1000 to 2500 ft. in length. The sluice closed in 0.03
second, and the fluctuations of pressure were automatically
registered. The maximum excess pressure due to water-hammer action was
as follows:--

+---------------------------------+---------------------------------+
| Pipe 4-in. diameter. | Pipe 6-in. diameter. |
+--------------+------------------+--------------+------------------+
| Velocity | Excess Pressure. | Velocity | Excess Pressure. |
| ft. per sec. | lb. per sq. in. | ft. per sec. | lb. per sq. in. |
+--------------+------------------+--------------+------------------+
| 0.5 | 31 | 0.6 | 43 |
| 2.9 | 168 | 3.0 | 173 |
| 4.1 | 232 | 5.6 | 369 |
| 9.2 | 519 | 7.5 | 426 |
+--------------+------------------+--------------+------------------+

In some cases, in fixing the thickness of water mains, 100 lb. per sq.
in. excess pressure is allowed to cover the effect of water hammer.
With the velocities usual in water mains, especially as no valves can
be quite suddenly closed, this appears to be a reasonable allowance
(see also Carpenter, _Am. Soc. Mech. Eng._, 1893).