Hydraulic Fluid Power. Andrea Vacca

Hydraulic Fluid Power - Andrea Vacca


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target="_blank" rel="nofollow" href="#ulink_47a7565f-609c-55f8-b2e7-9701108f5e10">Figure 3.11 clarifies how to identify the different loss terms in a pipe flow, between two reference sections 1 and 2. After every geometrical discontinuity in the piping system, such as at the entrance or after an elbow, or a fitting, the flow requires to travel a certain length before reaching fully developed conditions. Fully developed conditions are defined when the same velocity profile is held throughout the entire length of the constant area pipe. Figure 3.11 shows the entrance region of the flow getting into the first section of the pipe (length L1) from the tank. After the entrance region, where the flow profile is still developing, the velocity profile is constant until the 180° bend is reached. The flow at the exit of the bend enters a second constant sectional portion (length L2) and develops along a certain travel length before reaching fully developed conditions.

      A detailed description of the entrance or the developing flow region is outside the scope of this chapter, but it has been a topic of interest in many fluid mechanics problems. Hence, it is important for the reader to understand the typical approach used in pipe flow problems to describe the energy loss associated with different portions of the pipe system.

      3.5.2 Major Losses

      For the regions of fully developed flow, it is possible to analytically demonstrate that for laminar flow conditions:

Schematic illustration of an example of analysis of major and minor losses in a pipe flow.

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

      (3.30)f equals f left-parenthesis italic Re comma StartFraction e Over upper D Subscript h Baseline EndFraction right-parenthesis

      (3.31)StartFraction 1 Over StartRoot f EndRoot EndFraction equals minus 2 log left-parenthesis StartFraction e slash upper D Over 3.7 EndFraction plus StartFraction 2.51 Over Re StartRoot f EndRoot EndFraction right-parenthesis

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

      3.5.3 Minor Losses

      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:

      or

      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, kexit = 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 Le, which corresponds to the length of a straight pipe that would provide the same head loss.

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


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