Question 4.17: Problem: Air at 27 °C flows through a room heating duct with...
Problem: Air at 27 °C flows through a room heating duct with a volume flow rate of 0.5 m³/s. If the duct has a square cross‐section with each side 30 cm long, what are the velocity and mass flow rate of air in the duct?
Find: Velocity V and mass flow rate \dot{m} of air.
Known: Air temperature T = 27 °C = 300.15 K, air volume flow rate \dot{V} = 0.5 m³/s, duct cross‐ section 0.3 × 0.3 m.
Assumptions: Air is an ideal gas with gas constant R = 0.2870 kJ / kgK, atmospheric pressure is 101.325 kPa.
Governing Equation:
Mass flow rate \dot{m} = \rho A\pmb{V}
\rho =\frac{m}{V} =\frac{P}{RT} =\frac{101.325 \times 10^3 \ Pa}{0.2870 \times 10^3 \ J/kgK \times (273.15 + 27) \ K} =1.1762 \ kg/m^3
\dot{m} =\rho \dot{V} =1.1762 \ kg/m^3 \times 0.5 \ m^3/s=0.5881 \ kg/s
\pmb{V}=\frac{\dot{m} }{\rho A} =\frac{0.5881 \ kg/s}{1.1762 \ kg/m^3 \times (0.3 \ m)^2} =5.5556 \ m/s
Answer: The mass flow rate is 1.18 kg / m³ and the air velocity is 5.56 m / s.