the derivation of the basic equation, which will be used throughout this book for describing the motion of the piston of a hydraulic cylinder.
In Figure 3.8, the cylinder is illustrated through its ISO symbol. Let us consider the case of cylinder extension, against a resistive force F. The fluid enters the piston chamber, causing the piston to move with a certain velocity
(3.15)
(3.16)
where D and d are respectively the piston diameter and the rod diameter.
Figure 3.7 Conservation of mass in a hydraulic junction.
Figure 3.8 Representation of a hydraulic cylinder during the extension.
The conservation of mass (Eq. (3.11)) can be applied to CV1, which includes all the fluid at the piston chamber. The CV increases its size during the motion of the piston; therefore, the change in volume (first term of the conservation of mass) cannot be assumed to be zero. Considering the fluid density constant throughout CV1,
(3.17)
The second term of Eq. (3.11) can be written with a simple expression considering that there is only one opening section in the CV1's control surface:
(3.18)
With these simplifications, and assuming constant fluid density, Eq. (3.11) applied to CV1 (bore side CV) becomes
(3.19)
A similar expression can be derived by applying the conservation of mass to the rod side CV, CV2:
(3.20)
From the results obtained for each CV, CV1 and CV2, a relation between the flow rates at the two cylinder ports can be derived:
Equation (3.21) is valid for both cases of extension and retraction of the linear actuator, and essentially it shows that the geometrical area ratio of the cylinder corresponds to the ratio of the flow rates at the two cylinder work ports.
It is important to notice that for a given flow rate entering the cylinder, the external load F applied to the piston does not have an impact on the cylinder motion. This is true when fluid compressibility effects can be ignored, as in most of the typical hydraulic control systems, the external load will instead have a direct impact on the fluid pressure inside the cylinder chambers.
3.5 Bernoulli's Equation
Bernoulli's equation is one of the most important equations in fluid mechanics.
Bernoulli's equation establishes the concept of energy conservation within a flow.
In textbooks on basic fluid mechanics, Bernoulli's equation is derived using two possible methods: one considers the conservation of mass and the momentum equation applied to a differential CV and another – perhaps more intuitively – starts from the principle of energy conservation. The reader can refer to [15] for more details. In this chapter, the classic form of the Bernoulli's equation is presented without discussing its derivation, in order to highlight its implications in hydraulic systems:
This equation is valid under steady‐state conditions, for incompressible and inviscid (frictionless) flows. Each term of the equation has units of energy per unit mass (J/kg) and summarizes three possible ways in which a fluid can store energy:
(3.23)
Equation (3.22) is used to describe the relation between pressure and fluid velocity in a flow stream:
As previously discussed, in hydraulic systems, the operating pressure is so high that even large differences in elevation are mostly negligible in fluid power machines. However, large variations in fluid velocities and in pressure can be found within hydraulic components.
