Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 55

A hydraulic circuit comprises a network of components such as pumps, valves, cylinders, and filters, which are connected through fluid conveyance elements, such as hoses or pipes. The flow rate through the connecting hoses or pipes is a recurring parameter used in systems analysis. This section provides a high‐level review of the concept of flow rate starting from the generic case of flow through a pipe, represented in Figure 3.4.

Before entering into more specific considerations for the hydraulic case, it is important to recall the definition of scalar product. Figure 3.4 shows the generic case of a pipe where a particle of fluid is traveling with a defined velocity along the direction of the pipe. The figure also represents a generic surface area defined by the section of the pipe with a generic plane. As shown in the figure, both are vector quantities: vector denotes velocity, while vector has the magnitude of the area A and it is perpendicular to the plane defining it. Their scalar product is

(3.6)

From the scalar product of Eq. (3.6), the convention for the flow crossing a surface can be defined: an in‐flow is associated with a negative scalar product, while an out‐flow has a positive scalar product.

Figure 3.4 Flow through a pipe: velocity vector and surface area vector. (a) outflow; (b) inflow.

The scalar product in Eq. (3.6) is particularly important for the definition of volumetric flow rate across a section. In general:

(3.7)

Figure 3.4 represents the case of a single fluid particle, Eq. (3.7) integrates this relationship to the whole section of the pipe. The velocity of each particle on the section is different and its profile depends on the flow regime. For laminar flow conditions, where viscous effects prevail over fluid inertia effect, the velocity profile resembles the parabolic distribution illustrated in Figure 3.5a.

For turbulent flow conditions (Figure 3.5b), the velocity profiles are fuller in the middle of the pipe while still satisfying the no‐slip condition (zero velocity in proximity of the wall). The dimensionless Reynolds number is typically used to distinguish the flow regime conditions:

(3.8)

where, in Eq. (3.8), D_h represents the hydraulic diameter of the flow section:

(3.9)

A is the perpendicular cross‐sectional area of the pipe, while P_w 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

(3.10)

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, v_avg.

The v_avg 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 v_avg.

The average velocity v_avg 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.