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§ 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 Darcy’s Experiments on Friction in Pipes.—All previous experiments on the resistance of pipes were superseded by the remarkable researches carried out by H. P. G. Darcy (1803–1858), the Inspector-General of the Paris water works.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 11/2 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 = α1 simply, where α1 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;

ζ = α (1 + β/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)

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

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

ζ = (α + α1/d) + (β + β1/d2) /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
α1= .0003992
β= .0010182
β1= .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/4l = mVn,

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 = mvn/dx, 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 (vn/d3−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