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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 dpζG (χ/Ω) Q (v2/2g) dl = 0.

Dividing by GQ,

dz + dp/G + ζ (χ/Ω) (v2/2g) dl = 0.


z + p/G + ζ (χ/Ω) (v2/2g) l = constant.

§ 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 + ζ (χ/Ω) (v2/2g) l1 = z2 + p2/G + ζ (χ/Ω) (v2/2g) l2;

or, if l2l1 = L, rearranging the terms,

ζv2/2g = (1/L) {(z1 + p1/G) − (z2 + p2/G)} Ω/χ.
Fig. 81.

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 Ω/χ which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.

Introducing these values,

ζv2/2g = mh/L = mi.

For pipes of circular section, and diameter d,

m = Ω/χ = 1/4πd2/πd = 1/4d.


ζv2/2g = 1/4dh/L = 1/4 di;


h = ζ (4L/d) (v2/2g);

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 4ζL/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.

Fig. 82.

§ 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 ζ0v2/2g, corresponding to an expenditure of GQζ0v2/2g 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 ζ(4L/d) (v2/2g) corresponding to GQζ (4L/d) (v2/2g) foot-pounds of work. Hence

GQh = GQv2/2g + GQζ0v2/2g + GQζ·4L·v2/d·2g;

h = (1 + ζ0 + ζ·4L/d) v2/2g. v = 8.025 √ [hd / {(1 + ζ0)d + 4ζL} ].


If the pipe is bell-mouthed, ζ0 is about = .08. If the entrance to the pipe is cylindrical, ζ0 = 0.505. Hence 1 + ζ0 = 1.08 to 1.505. In general this is so small compared with ζ4L/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 41/2 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 ζ as mentioned below. Then from that value of d find a corrected value for ζ and repeat the calculation.

The equation above may be put in the form

h = (4ζ/d) [{ (1 + ζ0) d/4ζ} + L] v2/2g;

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

(1 + ζ0) d/4ζ

of the pipe. Putting 1 + ζ0 = 1.505 and ζ = 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 ζ for ordinary iron pipes is

ζ = 0.007567.

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 ζ on the velocity. Having regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that ζ might be expressed in the form

ζ = a + β/v.

The following are the best numerical values obtained for ζ so expressed:—

  α β
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

4ζ = α + β/√v = 0.003598 + 0.004289/√v.