rho EndFraction plus StartFraction alpha 1 v 1 squared minus alpha 2 v 2 squared Over 2 EndFraction plus g left-parenthesis z 1 minus z 1 right-parenthesis"/>
Equation (3.34) highlights the pressure, kinetics, and elevation terms contributing to hloss. In hydraulic systems, it is very common to have the pressure differential term dominating the other two, which can easily be ignored. For this reason, it can often be assumed that
(3.35)
In hydraulic systems, the head loss term hl (= hmajor + hminor) relates to a pressure loss.
Equations (3.25), (3.29) (major losses), and (3.32) (minor losses) highlight how the pressure drop due to frictional losses across a hydraulic element or a section of a pipe is proportional to v2, for turbulent conditions:
Common components of a hydraulic system such as valves and hydraulic fittings are mostly characterized by turbulent flow conditions. This is common for all sources of minor losses. Considering the relation between average velocity and volumetric flow rate (Eq. (3.10)), Eq. (3.36) can be written as follows:
(3.37)
The constant of proportionality in the quadratic equation between pressure can be referred to as turbulent hydraulic resistance, Rturb. The resistance term originates from the electric analogy approach. The value of the resistance sets the link between the generalized flow variable (electric current, in the electric domain, and volume flow rate, in the hydraulic domain) and the generalized effort variable (voltage, in the electric domain and pressure, in the hydraulic domain)2:
The hydraulic resistance R can be analytically derived from Eq. (3.29) (major losses) and Eq. (3.32) (minor losses), if the empirical friction factor or the loss coefficient is known. However, it is quite common for hydraulic components to find flow–pressure drop curves such as the one in Figure 3.14 (valid for a check valve), from which the resistance coefficient can be easily derived.
At this point, the reader should notice one major difference between the hydraulic and electrical resistances. In fact, in the hydraulic domain, the law is quadratic, while in the electrical one (Ohm's law), it is linear. This is because of the turbulent flow condition. The hydraulic–electrical analogy is completely accurate only for laminar flow conditions:
where Rlam is the laminar hydraulic resistance. The subscript designation of the hydraulic resistance R is different between Eqs. (3.38) and (3.39) and refers to the type of flow inside the component. In hydraulic applications, laminar flow occurs only in particular cases, such as leakage flows inside the small gaps of pumps or spool valves.
The hydraulic resistance expresses the relation between flow rate Q and pressure drop Δp across a hydraulic element. For laminar flow conditions, the hydraulic resistance Rlam is a constant of proportionality between Q and Δp. In the more common case of turbulent conditions, the hydraulic resistance is a coefficient between Q2 and Δp.
For simplified studies, or when linear relations are more convenient for applying control laws, it is possible to assume a linear relation between flow rate and pressure also for turbulent flow. This approximation is not recommended for general analysis, but it can be accurate enough to describe relative variations of pressure and flows in a small interval, as it is illustrated in Figure 3.15.
Figure 3.14 Resistance across a hydraulic check valve.
Figure 3.15 Linear approximation for the hydraulic resistance in turbulent flow conditions.
The linear hydraulic resistance (Rlin) can be calculated as
(3.40)
3.7 Stationary Modeling of Flow Networks
The following chapters of the book focus on the analysis of hydraulic systems operating in steady‐state conditions. Hence, after the presentation of the basic equations for hydraulic resistance and conservation of mass, it is now appropriate to provide the reader with the general approach that can be used to model a flow network.
A flow network can be defined as any collection of elements (valves, cylinders) and sources (pumps). The network interconnections are the fluid conveyance elements.
According to the approach also presented by Merritt [32], the flow and the pressure distribution within a network must satisfy three constraints:
1 Flow–pressure relationshipEvery element of the circuit is characterized by a flow–pressure relationship. The simplest example is the case of the hydraulic resistance that can be used to describe pipes, fittings, and certain hydraulic valves, previously shown in
