17. Mead losses (general considerations)

Losses of energy; or head losses fis they are commonly called; de­pend on the shape, size and roughness of a channel, the velocity and viscosity of a fluid; they do not depend on the absolute pres­sure of the fluid. Viscosity alone, though basically the primary cause of all head losses, hardly pver accounts for any substantial loss of energy. This, will be examined in detail further on.

Experiments show that in many cases head losses are approxi­mately proportional to the square of the velocity. Very long ago this prompted the following genera) expression for head losses;

(4.17)

and for pressure losses:

The convenience of this expression is that it contains a dimen-sionless coefficient of proportionality ξ, called the loss coefficient, and the velocity head from Bernoulli's equation. The loss coeffi­cient represents the ratio of head losses to velocity head.

Head losses are commonly divided into local losses (also form or minor losses) and friction (or major) losses.

Local losses are caused by local features, or so-called hydraulic resistances, such as change of shape or size of a channel. When a Quid flows through such local features its velocity changes and eddies may appear. Examples of local loss features are shown in Pig. 30: (a) gate, (b) orifice, (c) elbow, (d) valve. Local losses are determined from Eq. (4.17):

(4.17`)

or

which is known as Weisbacb's formula.

Here v is the mean velocity across the pipe section for which the local loss was deter­mined.* If the diameter, and hence the velocity of flow, changes along the length of a pipe, it is more convenient to take the highest velocity (i.e., the velocity through the nar­rowest portion) as the basis for calculation. Every local feature is characterised by a certain loss coefficient ξ which can, in many cases, be regarded as approximately constant for a given type of feature. Local losses will be further considered in Chapter VIII.

Friction, or major, losses are energy losses which develop in straight pipes of uniform cross-section, i. e., with uniform flow, and they increase as the length of the pipe (Fig. 31). These losses are due to the internal friction in a fluid and therefore they appear not only in rough pipes but in smooth pipes as well.

Loss of head due to friction can be expressed according to the general equation of head losses (4.17), i. e.,

(4.18)

A more convenient form, however, is one in which the loss coef­ficient ξ is correlated with the relative length of a pipe.

Consider a portion of circular pipe equal in length to its diameter, denoting the loss coefficient in Eq. (4.18) by the symbol λ. Then for the whole pipe of length l and diameter d (see Fig. 31) the loss coef­ficient will be times greater, i. e.,

and instead of Eq. (4.18) we have

(4.18`)

or

(4.19)

Equation (4.18') is commonly known as Darcy's formula.

The diqaenaionless coefficient X is called the friction factor. It represents a coefficient of proportionality between the friction losses, on the one hand, and the product of relative pipe length and velo­city head, on the other.

The physical meaning of A, is easily found from a consideration of the condition for uniform flow in a cylindrical pipe of length l and diameter d (see Fig. 31), namely that the sum of the two forces acting on the volume—pressure and friction—be zero. This equation has the form

where т, is the shear stress at the pipe wall.

From this, and taking into account Eq. (4.19), we readily obtain

(4.20)

i. e., the friction factor % is proportional to the ratio of the shear stress at the pipe wall and the dynamic pressure computed according to the mean velocity.

Head losses in closed conduits (pipes) in which a liquid is com­pletely enclosed by solid boundaries (i.e., has no free surface) are due to the specific potential energy decreasing along the flow. In this case, if the specific kinetic energy does change at the given rate of discharge, this is due not to energy losses but to a change in the cross-section of the channel, as kinetic energy is a function of velocity only, which in turn depends on the rate of dis­charge and. cross-sectional area:

Consequently, in a pipe of uniform cross-section the velocity and specific kinetic energy of a liquid are constant, irrespective of local disturbances and head losses. Loss of head in this case is determined as the difference between the level of the liquid in two open piezo­meter tubes (see Figs 30 and 31).

Computation of head losses for various specific cases is one of the fundamental problems of hydraulics, and the two following chap­ters will be devoted to this.

* Henceforth the subscript m will be used for v only when confusion is possible between mean velocity and local velocity.