§ 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experiments on the resistance of pipes were superseded by the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 / |
+--------------------+---------------+----------+---------------+
[Illustration: FIG. 83.]
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