[(bar · min)/l].
Find:
1 The motion of the cylinder in rest condition (vertical velocity)
2 The reading of the pressure gages A and B
Solution:
The hydraulic schematic of the lifting system considered in this problem introduces the symbolic representation of components that will be described in more detail in the next chapter of this book. However, for the reader, the functioning of the system should be quite intuitive: a hydraulic pump P is driven by a prime mover (an electric motor) M to supply flow to a hydraulic control valve, DCV. The hydraulic control valve is manually activated, and it determines the direction of the supply flow: in one position, it implements the cylinder extension, and in the other, the retraction. Centering springs ensure the valve blocks the cylinder work ports in case of no valve actuation.
A pressure relief valve is installed at the pump outlet, to protect the pump from overpressurization.
Considering the rest position, the supply part of the hydraulic system is isolated through the DCV, so that the problem can be simplified by considering the following schematic:
1 If a motion of the piston would occur in rest conditions the steady‐state form of the conservation of mass at both Vpist and Vrod would apply:(E4.4.1) (E4.4.2) The above equations were derived in Chapter 3, when analyzing the basic functioning of a linear actuator. The flow QLEAK is the flow rate through the laminar orifice that represents the internal leakages inside the cylinder.(E4.4.3) From the above (E4.4.1) and (E4.4.2), being Apist ≠ Arod, it is clear that the only possible solution for the leakage flow is QLEAK = 0 l/min. This means that even if there is an opening between the two chambers of a differential cylinder actuator, there are no internal leakage inside the actuator. This means that v = 0 m/s, meaning that the piston will remain in standstill.
2 The above Equation (E4.4.3) implies thatRecognizing that prod = pA and ppist = pB, we haveThe two pressure gages will give the same reading, which can be determined by the force balance of the piston (assuming no friction):
Further considerations:
This example shows how the presence of internal leakages paths inside a differential cylinder do not prevent realizing a perfect load holding, as long as the hydraulic circuit properly blocks the cylinder work ports. This is why load handling hydraulic machines can safely use even worn cylinders, as long as the connecting hydraulic valves can achieve a perfect sealing.
The answer is true only after the system reaches a stationary condition. A very fast transient will actually happen when the valve is brought to rest, or in case of a variation of the load. During this transient, the pressure inside the piston and rod chamber will equalize. This process involves a small mass exchange between Vpist and Vrod (therefore a QLEAK ≠ 0 for a short interval) related to the compressibility of the fluid.
Problems
1 4.1 An orifice is used in a system to discharge pressure to the tank from a hydraulic line. The pressure in the line (upstream the orifice) can reach 190 bar (gage pressure) during the operation of the system, while the tank pressure can be considered constant at 0 bar (gage pressure). If the diameter of the orifice is D = 0.5 mm, evaluate the flow rate through the orifice at the maximum pressure condition, as well as the energy dissipated through it. Assume the oil viscosity ρ = 850 kg/m3 and the orifice coefficient Cf = 0.62.
2 4.2 An orifice in parallel with a check valve is used to control the velocity of an actuator during its lowering, as shown in the figure below. Determine the diameter of the orifice to accomplish the complete lowering in 10 seconds starting from actuator fully extended. During lowering, the external load is such that the piston chamber is pressurized at constant pressure of 80 bar. The cylinder parameters are piston diameter = 63.5 mm, rod diameter 44.5 mm, and stroke 965 mm. The oil has a density of 880 kg/m3. The orifice coefficient is 0.65.
3 4.3 Consider again the system of the previous problem, but this time the orifice diameter is given dorif = 2 mm. What is the lowering time in this case?Introduce an additional orifice in series or in parallel to the given orifice to achieve a lowering time of 10 seconds. What is the diameter of this additional orifice?
4 4.4 Consider the system described in Example 4.4. With the same system data, consider now the case of a double rod cylinder instead of a single rod one. Will the system behave in the same way? What would be the pressure reading of the pressure gages in this case?
5 4.5 Consider again the problem described in Example 4.4. The system is brought to rest immediately after the raising of the 10 000 kg load by switching the position of the valve DCV. The piston will interrupt abruptly its upward motion and stop. Will the piston remain in standstill exactly at the position it reached when the DCV was switched? Or it will end up remaining stand still at a different position? Motivate your answer. If oil properties are needed, refer to the data below:Density850 kg/m3Bulk modulus15 000 barKinematic viscosity40 cSt
6 4.6 A valve experiences a pressure drop of 100 psi (687 kPa) for a flow rate of 25 gpm (98.4 l/min). The fluid is hydraulic oil with a specific gravity (sg) of 0.90 (roil = sg rwater). Determine the capacity coefficient k (). Determine also the area of the equivalent orifice, assuming Cd = 0.65.
7 4.7 For the six cases (a) to (f) shown below, indicate if the orifice behaves as a metering orifice or a compensator. Hint: Always consider the pump as a flow generator. For this problem, the accumulator can be considered as a constant pressure source.
8 4.8 For the two systems in figure below:Determine if the orifice O operates as a metering or as a compensator (give separate answers for the two systems).Considering F = 1 kN, dpiston = 10 mm; dorifice = 1 mm for both systems; pacc = 200 bar for the accumulator circuit; n = 500 rpm and VDpump = 5 cm3/rev for the pump circuit, determine the extension speed of the cylinder for the two cases. You can assume Cf = 0.7, ρ = 800 kg/m3.If an additional 3 mm orifice is put in series to the orifice O, does the cylinder velocity will change for the two cases?
Notes
1 1 It should be noticed that the velocity profile approaches uniform flow only at high Reynolds numbers (turbulent conditions).
2 2 With commonly used metric or standard units:
Конец ознакомительного фрагмента.
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