Question

A turbojet is flying with a velocity of \(900 \mathrm{ft} / \mathrm{s}\) at an altitude of \(20,000 \mathrm{ft}\), where the ambient conditions are 7 psia and \(10^{\circ} \mathrm{F}\). The pressure ratio across the compressor is \(13,\) and the temperature at the turbine inlet is 2400 R. Assuming ideal operation for all components and constant specific heats for air at room temperature, determine ( \(a\) ) the pressure at the turbine exit, \((b)\) the velocity of the exhaust gases, and \((c)\) the propulsive efficiency.

Short answer

#Answer# (a) To find the pressure at the turbine exit, we can use the following steps: 1. Calculate the pressure at the compressor exit: \(P_\text{compressor} = \text{Pressure Ratio} \cdot P_\text{inlet} = 13 \cdot 7 = 91\) psia. 2. Calculate the temperature at the turbine exit using the relation: \(T_\text{turbine\_exit} = T_\text{turbine\_inlet} - \frac{w_\text{compressor}}{C_p}\). 3. Use the isentropic relationship to find \(P_\text{turbine\_exit}\). Calculate the specific values using the given information and equations. (b) To find the velocity of the exhaust gases, use the conservation of energy equation: \(v_\text{exhaust}^2 = v_\text{inlet}^2 + 2 C_p(T_\text{turbine\_inlet} - T_\text{turbine\_exit})\) Calculate the specific value using the given information and equation. (c) To find the propulsive efficiency, use the following formula: \(\eta_\text{propulsive} = \frac{2 \cdot \frac{v_\text{inlet}}{v_\text{exhaust}}}{1 + \frac{v_\text{inlet}}{v_\text{exhaust}}}\) Calculate the specific value using the given information and equation.

Step-by-step solution

Find the conditions at the inlet and exit of the compressor

Given the ambient pressure of 7 psia and altitude of 20,000 feet, we can find the temperature using the Ideal Gas Law: \(P_\text{inlet} = R \cdot T_\text{inlet} \cdot \rho_\text{inlet}\). We are given the pressure ratio across the compressor, \(\text{Pressure Ratio} = \frac{P_\text{compressor}}{P_\text{inlet}}\). By rearranging the expression, we can find the pressure at the exit of the compressor: \(P_\text{compressor} = \text{Pressure Ratio} \cdot P_\text{inlet} = 13 \cdot 7 = 91\) psia.

Find the conditions at the turbine inlet

We are given the temperature at the turbine inlet (2400 R). Since we also have the pressure at the compressor exit, which is the same as the turbine inlet, we have all necessary information about the turbine inlet.

Use the standard turbojet equations to find the conditions at the turbine exit

We'll be using the following relations: 1. Heat addition in the combustor: \(q_\text{combustor} = C_p(T_\text{turbine\_inlet} - T_\text{compressor\_exit})\) 2. Turbine work output: \(w_\text{turbine} = -C_p(T_\text{turbine\_exit} - T_\text{turbine\_inlet})\) 3. Work input to the compressor: \(w_\text{compressor} = C_p(T_\text{compressor\_exit} - T_\text{compressor\_inlet})\) By setting \(|w_\text{turbine}| = |w_\text{compressor}|\), the turbine work output equals the work input to the compressor. We can now find the exit temperature for turbine: \(T_\text{turbine\_exit} = T_\text{turbine\_inlet} - \frac{w_\text{compressor}}{C_p}\). We use the fact that both compressor and turbine are isentropic, so: \(\frac{P_\text{turbine\_exit}}{P_\text{turbine\_inlet}} = \left(\frac{T_\text{turbine\_exit}}{T_\text{turbine\_inlet}}\right)^{\frac{\gamma}{\gamma -1}}\). Rearrange and solve for \(P_\text{turbine\_exit}\) to obtain the pressure at the turbine exit. Answer to (a): Pressure at the turbine exit.

Calculate the velocity of the exhaust gases

To calculate the velocity of the exhaust gases, use the conservation of energy equation: \(v_\text{exhaust}^2 = v_\text{inlet}^2 + 2 C_p(T_\text{turbine\_inlet} - T_\text{turbine\_exit})\) By solving this equation, we can obtain the velocity of the exhaust gases. Answer to (b): Velocity of the exhaust gases.

Calculate the propulsive efficiency

The propulsive efficiency \(\eta_\text{propulsive}\) is given by the following formula: \(\eta_\text{propulsive} = \frac{2 \cdot \frac{v_\text{inlet}}{v_\text{exhaust}}}{1 + \frac{v_\text{inlet}}{v_\text{exhaust}}}\) By substituting the found values of \(v_\text{inlet}\) and \(v_\text{exhaust}\), we can calculate the propulsive efficiency. Answer to (c): Propulsive efficiency.