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Figure 3.11 Example of analysis of major and minor losses in a pipe flow.

The velocity value v represents the average velocity over the section. From considerations based on dimensional analysis, it is possible to derive the Darcy‐Weisbach equation, which has validity for both laminar and turbulent flow conditions:

(3.29)

      where the friction factor f is a function of the Reynolds number and the relative roughness of the pipe:

      (3.30)

Figure 3.12 shows the results from the pioneering work performed by Moody [15] on the determination of the friction factor. The Moody's diagram is nowadays the most used chart for determining the major head losses in pipe flows. Analytical expressions for f can also be found in basic fluid mechanics textbooks. A widely adopted one is the Colebrook's formula:

      (3.31)

From Eq. (3.28), it is possible to conclude that the loss term h_major in laminar conditions is proportional to the average fluid velocity (v) because of the presence of the Reynolds number, Re, at the denominator of the expression. However, under conditions of complete turbulence, where the friction factor f is constant, the loss term follows a quadratic relation with v.

      The major loss term h_major is proportional to the average fluid velocity, v, in laminar conditions, and to v² in complete turbulent conditions.

      3.5.3 Minor Losses

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

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