Question 7.4.10: Calculation of Inertance Consider fluid flow (either liquid ...

The mass of the slug is ρ AL, where ρ is the fluid mass density. The net force acting on the slug due to the pressures p_{1} and p_{2}  is A(p_{2}  −  p_{1}) . Applying Newton’s law to the slug, we have
ρ  AL \frac{d v}{d t} = A(p_{2}  −  p_{1})
where v is the fluid velocity. The velocity v is related to the mass flow rate q_m by ρ  Av = q_{m}.
Using this to substitute for v, we obtain
L \frac{d q_{m}}{d t} = A(p_{2}  −  p_{1})
or
\frac{L}{A} \frac{d q_{m}}{d t} = p_{2}  −  p_{1}

With p = p_{2}  −  p_{1}, we obtain
\frac{L}{A} = \frac{p}{d q_{m}/d t}
Thus, from the definition of inertance I,
I = \frac{L}{A}
Note that the inertance is larger for longer pipes and for pipes with smaller cross section.