Question

Air at \(7^{\circ} \mathrm{C}\) enters a turbojet engine at a rate of \(16 \mathrm{kg} / \mathrm{s}\) and at a velocity of \(300 \mathrm{m} / \mathrm{s}\) (relative to the engine). Air is heated in the combustion chamber at a rate \(15,000 \mathrm{kJ} / \mathrm{s}\) and it leaves the engine at \(427^{\circ} \mathrm{C}\). Determine the thrust produced by this turbojet engine. (Hint: Choose the entire engine as your control volume.

Short answer

Answer: The thrust produced by the turbojet engine is approximately 227.68 N.

Step-by-step solution

Determine the specific heat of air at constant pressure

Calculate the specific heat of air at constant pressure (cp) to be \((1005 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K})\). This value is commonly used for air and can be found in thermo-physical data tables.

Determine the temperature difference

Calculate the temperature difference between the inlet and outlet of the turbojet engine. \(\Delta T = T_{out} - T_{in} = (427 + 273) - (7 + 273) = 420 \mathrm{K}\), where we have converted temperatures to absolute values (Kelvin).

Calculate the change in velocity

Using the first law of thermodynamics, we know that the work done is equal to the heat added minus the kinetic energy change: \(W = Q - \Delta KE\). We are given the heat added \(Q = 15000 \mathrm{kJ} / \mathrm{s}\) and can calculate the change in kinetic energy as: \(\Delta KE = m\cdot(cp)\cdot\Delta{T} - W = (16 \mathrm{kg / s})\cdot(1005 \mathrm{J / kg \cdot K})\cdot(420 \mathrm{K}) - (15000 \cdot 1000 \mathrm{J / s}) = 1707600 \mathrm{J / s}\) Now, we determine the change in velocity: \(\Delta v = \Delta KE / (\frac{1}{2}mv) = 1707600 \mathrm{J / s} / (\frac{1}{2}(16 \mathrm{kg / s})(300 \mathrm{m / s})) = 14.23 \mathrm{m / s}\)

Calculate the exit velocity

Find the exit velocity of the air relative to the engine by adding the change in velocity to the initial velocity: \(v_{exit} = v_{in} + \Delta v = 300 \mathrm{m / s} + 14.23 \mathrm{m / s} = 314.23 \mathrm{m / s}\)

Determine the thrust produced by the engine

Finally, calculate the thrust produced by the engine using the following formula: \(F = m \cdot (v_{exit} - v_{in})\) \(F = (16 \mathrm{kg / s}) \cdot (314.23 \mathrm{m / s} - 300 \mathrm{m / s}) = 227.68 \mathrm{N}\) The thrust produced by this turbojet engine is approximately equal to 227.68 N.