alt="normal upper Omega Subscript e q comma p a r Baseline equals sigma-summation Underscript i Endscripts normal upper Omega Subscript i Baseline semicolon d Subscript e q comma p a r Baseline equals StartRoot sigma-summation Underscript i Endscripts d Subscript i Superscript 2 Baseline EndRoot"/>
For orifices in series, the flow across the orifices is constant, while the Δp at each orifice is different:
(4.14)
By definition, the equivalent orifice satisfies the equation:
(4.15)
By equating the last two equations, it is possible to provide the expression for Ωeq, ser. This can be shown for the case of two orifices below, assuming again the same flow coefficient for all orifices:
(4.16)
Therefore, the equivalent orifice area is
(4.17)
This equation can easily be generalized for more orifices:
(4.18)
Example 4.2 Orifice Sizing
An emergency supply system uses an accumulator pressurized at 100 bar to extend a cylinder. This has a piston diameter D = 15 cm and sees a force of 50 kN. The connection between the cylinder and the accumulator is managed through a normally closed solenoid valve. The valve, when energized, implements a restriction equivalent to an orifice (this feature is represented in the valve symbol) with diameter of 4.4 mm and flow coefficient of 0.7. Assuming the accumulator is large enough to maintain a constant supply pressure, calculate the cylinder extension speed. Elaborate a simple system modification (without changing any of the existing components) for reducing the extension velocity to 60% of the previous value. Assume the pressure of the accumulator constant during the cylinder extension.
Given:
The ISO schematic of the hydraulic emergency system, which includes an accumulator as energy source, an on/off valve, and a linear actuator. The upstream pressure of the accumulator, constant, pacc = 100 bar; the orifice diameter d0 = 4.4 mm and the flow coefficient Cf = 0.7. The load on the actuator is F = 50 kN and the piston diameter, D = 15 cm.
Find:
The extension velocity of the piston,
Solution:
The system contains a solenoid valve (SV; which will be further described in Chapter 8), which when energized opens an accumulator to the piston chamber of a linear accumulator. For simplicity, the accumulator can be seen as a constant pressure source. In reality, the pressure inside the accumulator will decrease as the accumulator releases flow, as it will be better described in Chapter 9.
When the valve is energized, the flow rate across it is defined by the orifice equation:
The pressure inside the cylinder is given by the external force:
Therefore, the actuator speed is
In order to reduce the actuator's speed, there are possible alternatives:
1 Increase
