- #1
LowEParticle
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This is a clearer, better drawn, and shorter version of a question I asked a few days ago. I apologize for not initially reducing it to this state - sometimes when reducing a question to its essence I stop too soon!
The diagram below shows two cross-sections of an axis-symmetric cylindrical chamber that has a valve armature passing through its center. The valve armature descends to close the chamber (left-hand drawing) and rises up to open the chamber (right-hand drawing). Where the valve armature's shaft passes through the bottom of the chamber it is sealed off by a frictionless, air-tight seal. The air pressure outside the chamber is always Phigh. If the chamber is closed, then the air pressure inside of it is Plow.
My problem is the total force Ftotal on the valve armature due to differential pressure when the chamber is closed. Ftotal is the sum of three terms:
F1 is the pressure inside of the chamber pressing upwards on the bottom of the valve armature plate (14mm radius):
F1 = Plow * [itex]\pi[/itex] * (14)2 (upwards)
F2 is the pressure outside of the chamber pressing upwards on the very bottom of the valve armature shaft (4mm radius):
F2 = Phigh * [itex]\pi[/itex] * (4)2 (upwards)
F3 is the pressure outside of the chamber pressing downwards on the top of the valve armature plate (18mm radius):
F3 = Phigh * [itex]\pi[/itex] * (18)2 (downwards)
So Ftotal = -F1 - F2 + F3
I showed all the significant radii on the right-hand drawing for easy identification, however, the solution above does not use the 20mm, 12mm, or 8mm radii. I have two questions:
Thank you very much for reading this and considering my problem.
Dave
The diagram below shows two cross-sections of an axis-symmetric cylindrical chamber that has a valve armature passing through its center. The valve armature descends to close the chamber (left-hand drawing) and rises up to open the chamber (right-hand drawing). Where the valve armature's shaft passes through the bottom of the chamber it is sealed off by a frictionless, air-tight seal. The air pressure outside the chamber is always Phigh. If the chamber is closed, then the air pressure inside of it is Plow.
My problem is the total force Ftotal on the valve armature due to differential pressure when the chamber is closed. Ftotal is the sum of three terms:
F1 is the pressure inside of the chamber pressing upwards on the bottom of the valve armature plate (14mm radius):
F1 = Plow * [itex]\pi[/itex] * (14)2 (upwards)
F2 is the pressure outside of the chamber pressing upwards on the very bottom of the valve armature shaft (4mm radius):
F2 = Phigh * [itex]\pi[/itex] * (4)2 (upwards)
F3 is the pressure outside of the chamber pressing downwards on the top of the valve armature plate (18mm radius):
F3 = Phigh * [itex]\pi[/itex] * (18)2 (downwards)
So Ftotal = -F1 - F2 + F3
I showed all the significant radii on the right-hand drawing for easy identification, however, the solution above does not use the 20mm, 12mm, or 8mm radii. I have two questions:
- Have I picked the correct radii to correctly calculate the relevant forces?
- Are the 3 forces I've identified the only ones produced by differential pressures that are relevant to the caclulation of Ftotal?
Thank you very much for reading this and considering my problem.
Dave