Question

A turbojet aircraft is flying with a velocity of \(280 \mathrm{m} / \mathrm{s}\) at an altitude of \(9150 \mathrm{m},\) where the ambient conditions are \(32 \mathrm{kPa}\) and \(-32^{\circ} \mathrm{C} .\) The pressure ratio across the compressor is \(12,\) and the temperature at the turbine inlet is 1100 K. Air enters the compressor at a rate of \(50 \mathrm{kg} / \mathrm{s}\), and the jet fuel has a heating value of \(42,700 \mathrm{kJ} / \mathrm{kg}\). Assuming ideal operation for all components and constant specific heats for air at room temperature, determine ( \(a\) ) the velocity of the exhaust gases, \((b)\) the propulsive power developed, and \((c)\) the rate of fuel consumption.

Short answer

In this problem, we are asked to determine the exhaust gas velocity, propulsive power developed, and fuel consumption rate of a turbojet aircraft in given ambient conditions. By making use of the ideal Brayton cycle and the associated formulas, we can follow these steps: 1. Calculate the air properties (enthalpy and temperature) at the given altitude 2. Calculate the compressor exit properties (enthalpy and temperature) 3. Calculate the heat addition in the combustor 4. Determine the rate of fuel consumption 5. Determine the exhaust gas velocity 6. Determine the propulsive power developed By following these steps and using the given information, we can successfully find the exhaust gas velocity, propulsive power developed, and the rate of fuel consumption for the turbojet aircraft.

Step-by-step solution

Calculate the air properties at the altitude

Since air has constant specific heats at room temperature, we can use the following relations: 1. For the isentropic process: \(\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma -1)/\gamma}\) 2. For the constant specific heat: \(c_p(T_2-T_1) = H_{2} - H_{1}\) where \(T_1\) is the temperature at sea level, \(P_1\) is the pressure at sea level, \(T_2\) is the temperature at the given altitude, \(P_2\) is the pressure at the given altitude, \(\gamma\) is the specific heat ratio of air (\(\approx 1.4\) for air), \(c_p\) is the specific heat at constant pressure, \(H_{1}\) is the enthalpy of the air at sea level, and \(H_{2}\) is the enthalpy of the air at the given altitude. Using the given ambient conditions of \(32 kPa\) and \(-32^{\circ} C\), we can calculate \(T_1\) and \(P_1\). We have \(T_1 = (273 - 32) K = 241 K\) and \(P_1 = 32 kPa\).

Calculate compressor exit properties

We are given the pressure ratio across the compressor as \(12\). Therefore, we can calculate the temperature and enthalpy at the compressor exit (\(T_3, H_{3}\)) using the above relations: \(\frac{T_3}{T_1} = \left(\frac{P_3}{P_1}\right)^{(\gamma -1)/\gamma}\) Since \(P_3 = P_2 \times 12\), we can find the temperature at the compressor exit, \(T_3\). Then, we can find \(H_{3}\) using the relation: \(H_{3} = c_p(T_3-T_1) + H_{1}\)

Calculate the heat addition in the combustor

We are given that the temperature at the turbine inlet is \(1100 K\). Let's denote this temperature by \(T_4\), and denote the enthalpy at this point by \(H_4\). We can find \(H_4\) using the relation: \(H_4 = c_p(T_4-T_1) + H_{1}\) Now, we can calculate the heat addition in the combustor, \(q_{in}\), using the relation: \(q_{in} = H_4 - H_3\)

Determine the rate of fuel consumption

We are given that the jet fuel has a heating value of \(42,700 kJ/kg\). We can use this information to determine the rate of fuel consumption, \(m_{f}\), using the following relation: \(m_{f} = \frac{q_{in}}{42,700}\)

Determine the exhaust gas velocity

The exhaust gas velocity, \(V_{ex}\) can be determined by considering the energy balance across the turbine and the nozzle: \(m_{f}(H_5 - H_4) = \frac{1}{2} m_a (V_{ex}^2 - V_0^2)\) Where \(H_5\) is the enthalpy at the turbine exit, \(V_0\) is the aircraft velocity given as \(280 m/s\), and \(m_a\) is the mass flow rate of air, given as \(50 kg/s\). We can calculate the enthalpy at the turbine exit, \(H_5\) by considering isentropic expansion across the turbine: \(H_5 = H_4 - (H_3 - H_2)\) Now, we can solve for the exhaust gas velocity \(V_{ex}\).

Determine the propulsive power developed

Propulsive power developed, \(P_{prop}\), can be calculated using the following relation: \(P_{prop} = m_a(V_{ex} - V_0) \times V_0\) Now we have all the required values to find the exhaust gas velocity, propulsive power developed, and rate of fuel consumption.