Pascal's Principle — Question 7

Question 7

A trapped bubble consumes stroke before useful motion

A hydraulic line contains a trapped gas bubble of volume $V_0=2.0\,\mathrm{cm^3}$ at absolute pressure $p_0=100\,\mathrm{kPa}$. The rest of the liquid is incompressible. Before the output can move, line pressure must reach $500\,\mathrm{kPa}$ absolute. A master piston of area $A_m=1.0\,\mathrm{cm^2}$ supplies the volume change. Assume isothermal bubble compression, $pV=\text{constant}$.

1. Find the input travel needed solely to compress the bubble to the lifting threshold.

2. At that threshold, does the bubble prevent static pressure transmission? Explain the distinction between force transmission and displacement response.

Question 7 - Solution

A trapped bubble consumes stroke before useful motion

(a) Derivation and calculation

$$p_0V_0=p_fV_f\Rightarrow\boxed{V_f=\frac{100}{500}(2.0)=.400\,\mathrm{cm^3}}.$$$$\Delta V=V_0-V_f=1.60\,\mathrm{cm^3}.$$$$\boxed{x_m=\Delta V/A_m=1.60/1.0=1.60\,\mathrm{cm}}.$$

(b) Result and interpretation

At static equilibrium, gas and liquid share the local pressure. The volume-pressure response is$$V(p)=p_0V_0/p,\qquad-\frac{dV}{dp}=\frac{p_0V_0}{p^2}>0.$$$$A_mx_m=V_0-V(p)\quad\text{before output motion}.$$The bubble adds compliance and changes stroke and transient response, but does not prevent static pressure transmission.

Physical check

The compression law requires absolute pressure; gauge-pressure ratios would give the wrong bubble volume.