Question

Air (of density 0.5 kg/m\(^3\)) enters horizontally into a jet engine at a steady speed of 200 m/s through an inlet area of 1.0 m\(^2\). Upon entering the engine, the air passes through the combustion chamber and the exhaust gas exits the jet engine horizontally at a constant speed of 700 m/s. The fuel mass flow rate added in the combustion chamber is negligible compared to the air mass flow rate. Also neglect the pressure difference between the inlet air and the exhaust gas. The absolute value of the horizontal force (in kN, up to one decimal place) on the jet engine is \(\underline{\hspace{1cm}}\).

Hint:

To calculate the force on the jet engine, use the momentum equation \( F = \dot{m} \Delta v \) to account for the change in velocity.

Answer 1

Correct Answer: 48

Using the principle of momentum conservation, the horizontal force is given by the equation:
\[ F = \dot{m} (v_{\text{exit}} - v_{\text{inlet}}) \] Where:
- \( \dot{m} = \rho A v_{\text{inlet}} \) is the mass flow rate,
- \( v_{\text{inlet}} = 200 \, \text{m/s} \),
- \( v_{\text{exit}} = 700 \, \text{m/s} \),
- \( \rho = 0.5 \, \text{kg/m}^3 \) is the density,
- \( A = 1 \, \text{m}^2 \) is the inlet area.
Substituting the values, we first calculate the mass flow rate:
\[ \dot{m} = 0.5 \times 1 \times 200 = 100 \, \text{kg/s} \] Now, the horizontal force is:
\[ F = 100 \times (700 - 200) = 100 \times 500 = 50,000 \, \text{N} = 50 \, \text{kN}. \] Thus, the absolute value of the horizontal force is approximately \( 50.0 \, \text{kN} \).