Inertance

Inertance is a measure of the pressure gradient in a fluid required to cause a change in flow-rate with time. The unit of inertance is Pa m−3 s2 and the usual symbol is I.

For a circular tube the inertance is given by:

I = {\rho \cdot \ell \over A} \,

where

ρ is the density (in units of kg / m3) of the fluid
ℓ is the length (in units of meters) of the tube
A is the cross sectional area (in units of m2) of the tube

The pressure gradient is related to the change in flow-rate by the equation:

\Delta p = I \cdot {\dot Q} = I \cdot {dQ \over dt}

To some it may appear counterintuitive that an increase in cross sectional area of a tube reduces the inertance of the tube. However, for the same mass flow-rate, a lower cross sectional area implies a higher fluid velocity and therefore a higher pressure gradient to accelerate the fluid.

The equation assumes constant density, that all parts of the fluid undergo the same acceleration and that the flow is fully developed (turbulent) "plug flow". This precludes sharp bends, water hammer and so on.

In respiratory physiology Inertance (of air) is measured in cmH2O L−1 s2.

1 cmH2O L−1 s2 ≈ 98100 Pa m−3 s2.

Using small-signal analysis, an inertance can be represented as a fluid reactance through the relation:

X = j \omega I

where

\omega = 2 \pi f
f is the frequency in Hz.

References