href="#ulink_2dee393b-a09f-5be9-8d33-e96c7823ab5a">(3.8)where, in Eq. (3.8), Dh represents the hydraulic diameter of the flow section:
(3.9)
A is the perpendicular cross‐sectional area of the pipe, while Pw is the wetted perimeter. For circular pipes, the hydraulic diameter corresponds to the pipe's internal diameter.
Figure 3.5 Actual velocity profile (laminar and turbulent flow) and uniform velocity profile. (a) Laminar regime. (b) Turbulent regime. (c) Uniform assumption approximation.
For circular pipes, Re < 2300 corresponds to laminar flow conditions, while Re > 4000 to turbulent flow conditions [15]. A transitional flow region is further defined between these flow regimes.
The average velocity value used for the calculation of the Reynolds number is defined as
The concept of average velocity can be illustrated with the uniform flow distribution in Figure 3.5b.
The uniform flow distribution permits to describe the overall flow through a pipe section with a single value, which is also the representative fluid velocity, vavg.
The vavg will be used for deriving many features of pipe flows, particularly for describing phenomena with empirical correlations. A significant example, which will be further detailed in Section 3.5, is the case of the frictional losses in a pipe. These are calculated from the value of vavg.
The average velocity vavg is also often one of the main parameters to be considered when sizing certain hydraulic components. For example, when selecting the proper diameter of the pipes, or the diameter of the ports of pumps or motors, designers have to ensure that the maximum average velocity reached during the operation of the system is below certain values. Design guidelines usually recommend the following maximum values for average velocity [30]:
Pressure lines – 25 ft/s or 7.62 m/s
Return lines – 10 ft/s or 3.05 m/s
Suction lines – 4 ft/s or 1.22 m/s
However, one must keep in mind that even when such requirement is met, the actual maximum fluid velocity at the centerline of the pipe is significantly higher.
3.4 Conservation of Mass
In fluid mechanics, the fundamental laws that describe flow can be expressed for a control volume (CV), which is a volume fixed in space or moving with a certain velocity through which the fluid flows.
The CV formulation of the mass conservation principle in fluid mechanics can be expressed by the following equation:
The first term represents the rate of change of the mass in the CV, while the second term is the net rate of flux of mass across the bounding control surfaces (CS; Figure 3.6). Details on the derivation of Eq. (3.11) can be found in basic fluid mechanics textbooks [15].
In most hydraulics problems, it is convenient to assume incompressible flow, as well as uniform flow at each inflow or outflow section of the control surface, so that
(3.12)
Qi represents all the flow rates exiting (positive) or entering (negative) the CV through the permeable surfaces. For stationary problems, the CV does not vary over time; therefore, the conservation of mass equation simply becomes
Figure 3.6 Control volume (CV) and bounding control surface (CS).
A direct application of the above equation is the case of a hydraulic junction (Figure 3.7) – the overall flow entering the junction equals the flow leaving the junction:
With this analogy, the law that applies in the flow in Eq. (3.14) is equivalent to Kirchhoff's current law applied to the nodes of electric circuits. Junctions in most of the hydraulic circuits are described in the following chapters, and the law that applies in Eq. (3.13) will be used frequently in describing the operation of the system.
3.4.1 Application to a Hydraulic Cylinder
Hydraulic cylinders are basic elements of a hydraulic system, and their function will be described in Chapter 7. This section is aimed to illustrate how the conservation of mass influences
