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Science / Forces / Undergraduate / sfd60

Hydraulic Press: Model Fluid Compliance

Derive a small-signal pressure transient for a fixed-volume chamber.

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5 questions0m 0s

Prerequisites

Dimensional analysis, conservation laws, differential equations and elementary fluid mechanics.

Learn the skill

Use lumped uniform pressure, constant bulk modulus B, reference volume V and small compression. Hydraulic compliance C = V/B. Continuity is C dp/dt = Q_in - A v - Q_leak. These are mathematical models, not equipment specifications. Use diagrams and calculations only; never build or operate a press for this activity.

Worked example

For V = 0.002 m^3 and B = 1 x 10^9 Pa, C = 2 x 10^-12 m^3/Pa. With piston blocked, no leaks and Q_in = 1 x 10^-6 m^3/s, dp/dt = 500,000 Pa/s.

In the small-compression model, inlet flow enters a chamber with a blocked piston. With compliance 2 times 10 to the minus 12 cubic meters per pascal and flow 10 to the minus 6 cubic meters per second, pressure rises at 500,000 pascals per second.
In the small-compression model, inlet flow enters a chamber with a blocked piston. With compliance 2 times 10 to the minus 12 cubic meters per pascal and flow 10 to the minus 6 cubic meters per second, pressure rises at 500,000 pascals per second.
Question 1 What is compliance V/B for the example?
Question 2 With the piston blocked and no leaks, what is dp/dt?
Question 3 Starting at zero gauge pressure, what is p after 0.2 s in this model?
Question 4 What incremental stored energy is 0.5 C p^2 at that pressure?
Question 5 If A v equals Q_in and there is no leak, what is dp/dt?

Further inquiry

Derive the continuity equation from volume balance. Integrate the blocked-piston pressure ramp from zero gauge pressure, and derive stored energy by integrating p C dp. Then include constant leakage conductance L with Q_leak = Lp and find the time constant and steady pressure. Explain when constant compliance and uniform pressure cease to be defensible.

Review criteria

  • State units, signs, initial conditions and all simplifying assumptions.
  • Check the derived result against conservation and at least one limiting case.
  • Explain model limitations without treating a classroom calculation as a machine design.