Fluid Mechanics seventh edition by Frank M. White · Problem P3.27

Integral Relations for a Control Volume — Frank M. White

Physics — Fluid Mechanics · unsteady-continuity-control-volume

Problem

Consider a highly pressurized air tank at conditions

(p_0, \rho_0, T_0)

and volume

\mathcal{V}_0

. In Chap. 9 we will learn that, if the tank is allowed to exhaust to the atmosphere through a well-designed converging nozzle of exit area

A

, the outgoing mass flow rate will be

\dot{m} = \frac{\alpha\, p_0 A}{\sqrt{RT_0}} \quad \text{where } \alpha \approx 0.685 \text{ for air}

This rate persists as long as

p_0

is at least twice as large as the atmospheric pressure. Assuming constant

T_0

and an ideal gas, (a) derive a formula for the change of density

\rho_0(t)

within the tank. (b) Analyze the time

\Delta t

required for the density to decrease by 25 percent.