Air enters a jet engine through section 1 with an area of A1=0.5 m2 at a (relative) speedof 250 m/s. The pressure at the entrance is atmospheric pressure (approximately 105 Pa)and the temperature is 300 K. Inside the engine, the injected fuel mixes perfectly withair, with fuel to air mass ratio of 1:30. The enthalpy and the kinetic energy flow rate ofthe fuel can be neglected at the fuel inlet. The combustion releases a heat per unit mass(of air) of 200 kJ/kg, after which the mixture is expanded through a nozzle to the exitsection 2 whose area is A2= 0.4 m2. The product of combustion can be taken to havethe same properties of air (such as Cp=1.005 kJ/kg k, γ=1.4 and so on). The nozzleperfectly expands the flow such that at the exit 2, the pressure is atmospheric pressure.The temperature of the flow at the exit is 400 K. The figure below shows a roughschematic of the engine and the flow. Find the thrust generated by the engine
Question
Air enters a jet engine through section 1 with an area of A1=0.5 m2 at a (relative) speedof 250 m/s. The pressure at the entrance is atmospheric pressure (approximately 105 Pa)and the temperature is 300 K. Inside the engine, the injected fuel mixes perfectly withair, with fuel to air mass ratio of 1:30. The enthalpy and the kinetic energy flow rate ofthe fuel can be neglected at the fuel inlet. The combustion releases a heat per unit mass(of air) of 200 kJ/kg, after which the mixture is expanded through a nozzle to the exitsection 2 whose area is A2= 0.4 m2. The product of combustion can be taken to havethe same properties of air (such as Cp=1.005 kJ/kg k, γ=1.4 and so on). The nozzleperfectly expands the flow such that at the exit 2, the pressure is atmospheric pressure.The temperature of the flow at the exit is 400 K. The figure below shows a roughschematic of the engine and the flow. Find the thrust generated by the engine
Solution
To find the thrust generated by the engine, we need to use the principle of conservation of momentum. The thrust (F) is given by the difference in momentum of the air-fuel mixture between the exit and the entrance of the engine.
The momentum of the air-fuel mixture at the entrance is given by the product of the mass flow rate (m_dot1) and the velocity (V1). The mass flow rate can be calculated using the equation m_dot1 = rho1 * A1 * V1, where rho1 is the density of the air at the entrance, A1 is the area of the entrance, and V1 is the velocity of the air at the entrance.
The momentum of the air-fuel mixture at the exit is given by the product of the mass flow rate (m_dot2) and the velocity (V2). The mass flow rate can be calculated using the equation m_dot2 = rho2 * A2 * V2, where rho2 is the density of the air at the exit, A2 is the area of the exit, and V2 is the velocity of the air at the exit.
The thrust (F) is then given by the equation F = m_dot2 * V2 - m_dot1 * V1.
To calculate the values of m_dot1, m_dot2, V1, and V2, we need to use the given values for A1, A2, the atmospheric pressure, the temperature at the entrance and exit, and the properties of air.
The density of the air at the entrance and exit can be calculated using the ideal gas law, rho = P / (R * T), where P is the pressure, R is the specific gas constant, and T is the temperature.
The velocity of the air at the entrance is given as 250 m/s. The velocity of the air at the exit can be calculated using the equation V2 = sqrt(2 * Cp * (T1 - T2)), where Cp is the specific heat capacity at constant pressure, T1 is the temperature at the entrance, and T2 is the temperature at the exit.
After calculating the values of m_dot1, m_dot2, V1, and V2, we can substitute them into the equation for the thrust to find the thrust generated by the engine.
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