§ 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experiments on the resistance of pipes were superseded by the remarkable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 112 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
ζ = α + β/*v*; but in pipes which have been some time in use it is
sufficiently accurate to take ζ = α_{1} simply, where α_{1} depends on the
diameter of the pipe alone, but α and β 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;

*d*).

α | β | |

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 | ζ = .005 (1 + 1/12d) |

For slightly incrusted pipes | ζ = .01 (1 + 1/12d) |

*a*)

*Darcy’s Value of the Coefficient of Friction ζ for Velocities not less**than 4 in. per second.*

Diameter of Pipe in Inches. | ζ | Diameter of Pipe in Inches. | ζ | ||

New Pipes. | Incrusted Pipes. | New Pipes. | Incrusted 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 ζ are, however, not exact for widely differing velocities. To embrace all cases Darcy proposed the expression

_{1}/

*d*) + (β + β

_{1}/

*d*

^{2}) /

*v*;

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

α | = .004346 |

α_{1} | = .0003992 |

β | = .0010182 |

β_{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* √(*mi*), Ganguillet and Kutter take

c = | 41.6 + 1.811/n + .00281/i |

1 + [ (41.6 + .00281/i) (n/ √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 ζ
range from 0.16 to 0.0028 in different cases. Hence means of discriminating
the probable value of ζ are necessary in using the equations
for practical purposes. To a certain extent the knowledge that
ζ 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 ζ = .0051, and according to the other
ζ = .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 ζ 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*/4

*l*=

*m*V

^{n},

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

*m*+

*n*log

*v*= log (

*dh*/4

*l*),

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}) P2−*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.*

Surface of Pipe. | Authority. | Diameter of Pipe in Metres. | Values of n. | |

Tin plate | Bossut | .036 | 1.697 | 1.72 |

.054 | 1.730 | |||

Wrought iron (gas pipe) | Hamilton Smith | .0159 | 1.756 | 1.75 |

.0267 | 1.770 | |||

Lead | Darcy | .014 | 1.866 | 1.77 |

.027 | 1.755 | |||

.041 | 1.760 | |||

Clean brass | Mair | .036 | 1.795 | 1.795 |

Asphalted | Hamilton Smith | .0266 | 1.760 | 1.85 |

Lampe. | .4185 | 1.850 | ||

W. W. Bonn | .306 | 1.582 | ||

Stearns | 1.219 | 1.880 | ||

Riveted wrought iron | Hamilton Smith | .2776 | 1.804 | 1.87 |

.3219 | 1.892 | |||

.3749 | 1.852 | |||

Wrought iron (gas pipe) | Darcy | .0122 | 1.900 | 1.87 |

.0266 | 1.899 | |||

.0395 | 1.838 | |||

New cast iron | Darcy | .0819 | 1.950 | 1.95 |

.137 | 1.923 | |||

.188 | 1.957 | |||

.50 | 1.950 | |||

Cleaned cast iron | Darcy | .0364 | 1.835 | 2.00 |

.0801 | 2.000 | |||

.2447 | 2.000 | |||

.397 | 2.07 | |||

Incrusted cast iron | Darcy | .0359 | 1.980 | 2.00 |

.0795 | 1.990 | |||

.2432 | 1.990 |