The Equation of Continuity — Question 7

Question 7

Storage must be accounted for before declaring a leak

A rigid gas vessel of volume $V=0.50\,\mathrm{m^3}$ has measured inlet mass flow $0.080\,\mathrm{kg/s}$ and measured outlet mass flow $0.050\,\mathrm{kg/s}$. During a $20\,\mathrm{s}$ interval its spatially uniform density rises from $1.0$ to $1.8\,\mathrm{kg/m^3}$. Assume these rates are constant and an unmeasured leak, if present, is outward.

1. Find the mass accumulation and the average leak mass flow required by the data.

2. What density would be predicted at the end if there were no leak? Why is the inlet-minus-outlet difference alone not the leak rate?

Question 7 - Solution

Storage must be accounted for before declaring a leak

(a) Derivation and calculation

$$\boxed{\Delta M=V(\rho_f-\rho_i)=.500(1.80-1.00)=.400\,\mathrm{kg}}.$$$$\frac{\Delta M}{\Delta t}=.400/20=.0200\,\mathrm{kg/s}.$$$$\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}-\dot m_{\mathrm{leak}}=dM/dt.$$$$\boxed{\overline{\dot m}_{\mathrm{leak}}=.080-.050-.020=.0100\,\mathrm{kg/s}}.$$

(b) Result and interpretation

With no leak,$$\Delta M_0=(.080-.050)(20)=.600\,\mathrm{kg}.$$$$\boxed{\rho_{f,0}=\rho_i+\Delta M_0/V=1.00+.600/.500=2.20\,\mathrm{kg/m^3}}.$$The measured port difference supplies accumulation as well as any leak; it is not itself the leak rate.

Physical check

The measured final density is lower than the no-leak prediction, consistent with positive outward leakage.