#### Conservation of Mass: Example Problems

Try to solve these problems before watching the solutions in the screencasts.

##### Example Problem 1

Air flows steadily through this nozzle. The nozzle has an inlet diameter of \(D_1\) = 0.1 m and an outlet diameter of \(D_2\) = 0.04 m. The density of the air decreases from \(\rho_1\) = 1 kg/m^{3} at the inlet to \(\rho_2\) = 0.5 kg/m^{3} at the outlet. The average speed of the air at the inlet is \(V_1\) = 12 m/s.

1. What is the average speed of the air leaving the nozzle, \(V_2\), in units of m/s?

2. What is the mass flow rate of the air leaving the nozzle, \(\dot{m}_2\), in units of kg/s? Is it smaller than, equal to, or greater than the mass flow rate of air entering the nozzle, \(\dot{m}_1\)?

3. What is the volumetric flow rate of the air leaving the nozzle, \(Q_2\), in units of m^{3}/s? Is it smaller than, equal to, or greater than the volumetric flow rate of air entering the nozzle, \(Q_1\)?

##### Example Problem 2

Water flows steadily through a porous cylindrical filter. The feed enters the filter on the left. The permeate flows perpendicularly through the wall of the filter with a velocity \(v_r\), which decreases along the length of the filter. The fluid that does not permeate the filter exits the filter as retentate. The feed enters the filter at a volumetric flow rate of \(Q_F\) = 20 L/s. What are the volumetric flow rates of the permeate, \(Q_P\), and retentate, \(Q_R\)? What percentage of the feed leaves as permeate?

Length of filter: \(L\) = 0.95 m

Diameter of filter: \(D\) = 60 mm

\(v_r = V_0 \left[ 1 \, – \, \left( \frac{z}{L}\right)^2 \right]\)

\(V_0\) = 4 cm/s