accessories typically present in hydraulic reservoirs, not shown in Figure 2.19, include inspection ports, level indicators, filling inlet port and drainage or discharge caps, internal de‐aeration separators (which are used to promote air separation and reduce foam), and magnets to capture contamination. The bottom of the tank is usually concave so that the solid impurities that remain at the bottom of the tank can be periodically drained through the discharge caps.
Figure 2.19 Typical design of a hydraulic reservoir.
Problems
1 2.1 A hydraulic cylinder is loaded such that the pressure in the piston chamber rises from 0 to 4000 psi (this chamber being blocked, for example, by a normally closed valve). Evaluate the motion Δx of the piston, in inches, as effect of oil compressibility. Assume no air entrapped in oil and Bliq = 2.2 · 105 psi (bulk modulus). The rod chamber is connected to atmosphere at null pressure. Neglect compressibility of the material of the cylinder.
2 2.2 The cylinder in figure is loaded from 0 to 4000 psi keeping the bore port blocked. Evaluate the motion Δx of the piston from its initial position, in inches (in.), as effect of oil compressibility. Assume no air entrapped in oil and Bliq= 2.2 · 105 psi (bulk modulus). The rod chamber is connected to atmosphere at null pressure. Neglect the compressibility of the material of the cylinder.How much is the force F?
3 2.3 A cylinder with a 3 in bore and an extension of 1.42 in from the retracted position is loaded from 0 bar to 5000 psi while keeping the bore port blocked (as in the previous problem figure). Evaluate the motion Δx of the piston, as effect of oil compressibility. Assume 5% air entrained in oil and Kliq = 2.2 105 psi. Neglect the compressibility of the material and of the cylinder walls. When evaluating the bulk modulus of the air, consider following two cases:Gradual (slow) process (isothermal)Fast process (adiabatic)
4 2.4 Measurements are taken on the piston chamber of a cylinder with a bore diameter of 250 mm. The bore port is blocked, and the rod chamber is connected to atmosphere (as in the previous problem figure). When the stroke (from retracted position) is 225 mm, the pressure is 70 bar. When the stroke is reduced to 222 mm, the pressure is 140 bar.Determine the bulk modulus of the fluid.
Notes
1 1 In the British units, the viscosity is expressed as (lbf · s)/ft.
2 2 The polytrophic constant can be assumed as a function of the dynamic of the process. For a fast compression or expansion process, γ ≅ 1.4 (adiabatic); for a slow process, γ ≅ 1 (isothermal). Intermediate choices, such as γ = 1.2, are often made to describe a more general situation.
Chapter 3 Fundamental Equations
The fundamental principles of fluid power are closely related to those of fluid mechanics: the basic relations used to analyze the operation of a fluid power system originate from the fundamental equations of fluid mechanics.
The basic concepts of hydraulic systems are usually presented with the simplified assumption of stationary conditions. This allows deriving very simple equations for the working fluid, which are particularly suitable to describe the functioning of even complex system layout architectures. This chapter describes how these simplified equations are derived from the classic equations of fluid mechanics of general validity. Readers with basic knowledge of fluid mechanics will appreciate and understand the derivation of these basic equations used throughout this book.
Knowledge of the tools used to perform a dynamic analysis is not necessary to understand the basic operation of a hydraulic circuit. This dynamic analysis becomes necessary when studying the system behavior during transients, for example, a sudden variation of the external load or the commutation of the system from one state to another (such as the sudden change in the commanded position of a hydraulic control valve). Some basic concepts pertaining to the dynamic behavior of hydraulic control systems during transients will be introduced in Chapter 5.
3.1 Pascal's Law
Pascal's law states that the pressure is transmitted undiminished in a confined body of fluid at rest.
The Pascal's law of fluid statics is the foundation of what is considered by most engineers the modern era of fluid power technology.
The first hydraulic machines of the nineteenth century, such as the first hydraulic presses and hydraulic lifts, are based on this law.
The use of Pascal's law in an elementary hydraulic machine is shown in Figure 3.1. At level z*, the fluid pressure must be equal in both the vertical branches of the apparatus. Therefore:
It is evident that the geometrical ratio of the areas of the two pistons is related to the ratio of the force applied:
Figure 3.1 Basic hydraulic machine.
Equation (3.1) shows how it is possible to produce large loading forces using small geometrical areas and establishing high fluid pressures. This is based on power density, the main advantage of fluid power technology. Equation (3.2) is based on many hydraulic machines that require force multiplication, such as hydraulic brakes. For the practical applications of fluid power technology, the upper pressure limit (usually defined by the relief valve setting) is never given based on the fluid, but on the structural requirements of the components or of the machines. This also explains the current trend of increasing, where possible, the operating pressures of fluid power machines so that the power‐to‐weight ratio is reduced. A significant example for this is fluid power for aviation technology, where over the years the working pressure of the hydraulic actuation systems (flap and slat drives, landing gears, nose wheel steering, and many others) increased up to 210 bar, which is used in most commercial airliners. High performance military aircraft recently increased to 350 bar.
3.2 Basic Law of Fluid Statics
The Pascal's law presented in Section
