Question

Consider an aircraft powered by a turbojet engine that has a pressure ratio of \(9 .\) The aircraft is stationary on the ground, held in position by its brakes. The ambient air is at \(7^{\circ} \mathrm{C}\) and \(95 \mathrm{kPa}\) and enters the engine at a rate of \(20 \mathrm{kg} / \mathrm{s}\) The jet fuel has a heating value of \(42,700 \mathrm{kJ} / \mathrm{kg},\) and it is burned completely at a rate of \(0.5 \mathrm{kg} / \mathrm{s}\). Neglecting the effect of the diffuser and disregarding the slight increase in mass at the engine exit as well as the inefficiencies of engine components, determine the force that must be applied on the brakes to hold the plane stationary.

Short answer

Based on the provided information and calculations, the force that must be applied on the brakes to hold a stationary aircraft with a turbojet engine is approximately 925.6 N. This force is determined by calculating the change in velocity of the air-fuel mixture and then finding the change in momentum.

Step-by-step solution

Calculate the change in velocity

First, we'll find the total energy input by the fuel at rate of 0.5 kg/s, with a heating value of 42,700 kJ/kg. The total energy input per second (power) can be found by multiplying the mass flow rate of fuel and heating value: Total_energy_input = (0.5 kg/s) × (42,700 kJ/kg) = 21,350 kJ/s Next, we'll convert this energy to work done on the air, assuming all energy is converted to kinetic energy (ignoring potential energy and inefficiencies): Total_energy_input = (1/2) * Mass_flow_rate_air * Delta_v^2 Where Delta_v is the change in velocity of air To find Delta_v, rearrange the equation and solve for it: Delta_v^2 = (2 * Total_energy_input) / Mass_flow_rate_air Delta_v = sqrt((2 * 21,350 kJ/s) / (20 kg/s)) = sqrt(2,145 m^2/s^2) = 46.31 m/s

Calculate the required force

Now that we have the change in velocity, we can find the force required to hold the plane stationary. To do this, we'll use the following equation: Force = Mass_flow_rate * Delta_v Substitute the values into the equation: Force = (20 kg/s) * (46.31 m/s) = 925.6 N Thus, the force that must be applied on the brakes to hold the plane stationary is approximately 925.6 N.