The Equation of Continuity — Question 3

Question 3

A gas can gain volume flow without gaining mass flow

A steady gas stream enters a duct at density $\rho_1=2.0\,\mathrm{kg/m^3}$, area $A_1=0.010\,\mathrm{m^2}$ and mean speed $v_1=12\,\mathrm{m/s}$. It exits at density $\rho_2=1.2\,\mathrm{kg/m^3}$ and area $A_2=0.0060\,\mathrm{m^2}$. Density and speed are uniform across each section; there is no leak.

1. Find mass flow, exit speed and both volume flow rates.

2. What exit speed would the incompressible formula predict, and why is that wrong here? Does continuity alone explain the density change?

Question 3 - Solution

A gas can gain volume flow without gaining mass flow

(a) Derivation and calculation

$$\boxed{\dot m=\rho_1A_1v_1=(2.0)(.0100)(12)=.240\,\mathrm{kg/s}}.$$$$\boxed{v_2=\frac{\dot m}{\rho_2A_2}=\frac{.240}{1.20(.00600)}=33.3333\,\mathrm{m/s}}.$$$$\boxed{Q_1=A_1v_1=.120\,\mathrm{m^3/s}},\quad\boxed{Q_2=\dot m/\rho_2=.200\,\mathrm{m^3/s}}.$$

(b) Result and interpretation

Ignoring the density change would give$$v_{2,\mathrm{wrong}}=A_1v_1/A_2=.120/.00600=20.0\,\mathrm{m/s}.$$$$\rho_1Q_1=\rho_2Q_2,\qquad Q_2/Q_1=\rho_1/\rho_2=5/3.$$Mass flow is conserved while volume flow grows. Continuity alone does not supply the thermodynamics responsible for the density change.

Physical check

The correct exit mass flow is $1.20(.00600)(33.3333)=.240\,\mathrm{kg/s}$.