Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 61

Minor losses are primarily caused by flow separation effects. Figure 3.13 illustrates the case of the flow separation in proximity of a pipe entrance from a reservoir: the streamlines qualitatively illustrated the mixing areas of the separated zones, where energy is dissipated.

Flow separation effects such as the case in Figure 3.13 occur at every geometrical discontinuity of the pipe flow system. The energy losses in these cases are described by two alternative formulas:

(3.32)

or

(3.33)

As in the case of major losses, minor losses are quantified with respect to the kinetic term v²/2, by means of empirical relations based on experimental data. For many cases, particularly for entrances, exits, or sudden contractions or expansions, it is common to find in the literature the k coefficients. In the case of an exit to a tank, it is intuitive to consider that all the kinetic energy of the fluid inside the pipe will be dissipated; therefore, k_exit = 1. For other discontinuities, typically k < 1.

For other discontinuities, such as elbow or bends, it is more common to evaluate the friction coefficient f relative to the diameter representative of the discontinuity (i.e. the diameter of the curved pipe, for the case of an elbow) and use an empirical value of equivalent length L_e, which corresponds to the length of a straight pipe that would provide the same head loss.

Figure 3.12 Moody's diagram for the calculation of the friction factor.

Source: Moody's diagram, Darcy–Weisbach friction factor, wikipedia. Licensed under CC BY‐SA 4.0.

Figure 3.13 Example of separation region at a flow entrance.

A fluid power engineer must be ready to use both the formulas (3.32) and (3.33), depending on the data source that is available. For basic geometries, empirical data for minor losses can be found in the book of Idelchik [31].

It is important to point out that, in general, the minor losses include also the frictional effects of the developing flow region that follows the initial flow separation. This allows for a straightforward identification of the lengths to be used in the evaluation of the major loss terms. For example, with reference to the previous Figure 3.11, the entrance loss considers the additional losses associated to the entrance region in such a way that the evaluation of the major losses of the first straight section of the pipe is performed with the geometrical length L₁. Similarly, the lengths L₂ and L₃ will be used in Eq. (3.29) for the other major losses present between the reference sections.

In most cases, the minor loss term h_minor is proportional to the term v².