Question

Air enters the compressor of a regenerative gasturbine engine at \(310 \mathrm{K}\) and \(100 \mathrm{kPa}\), where it is compressed to \(900 \mathrm{kPa}\) and \(650 \mathrm{K}\). The regenerator has an effectiveness of 80 percent, and the air enters the turbine at 1400 K. For a turbine efficiency of 90 percent, determine \((a)\) the amount of heat transfer in the regenerator and ( \(b\) ) the thermal efficiency. Assume variable specific heats for air.

Short answer

Question: Determine the heat transfer in the regenerator and the thermal efficiency for a regenerative gas turbine engine, given the information provided. Answer: The heat transfer in the regenerator is c_p * 560 K, and the thermal efficiency of the gas turbine engine is 90%.

Step-by-step solution

Identify the given information and variables

In this exercise, we have the following information: - Inlet temperature: T1 = 310 K - Inlet pressure: P1 = 100 kPa - Outlet pressure from the compressor: P2 = 900 kPa - Outlet temperature from the compressor: T2 = 650 K - Regenerator effectiveness: 80% - Turbine inlet temperature: T3 = 1400 K - Turbine efficiency: 90% We need to find: (a) Heat transfer in the regenerator (b) Thermal efficiency of the gas turbine engine

Calculate the temperature after the regenerator

To calculate the temperature of the air after the regenerator, we can use the equation for the effectiveness of the regenerator (η_regeneration): η_regeneration = (T4 - T2) / (T3 - T2) Let's solve for T4 which is the temperature of the air after the regenerator: T4 = T2 + η_regeneration * (T3 - T2) Use the given values of η_regeneration (80%), T2 (650 K) and T3 (1400 K) to calculate T4: T4 = 650 + 0.8 * (1400 - 650) T4 = 1210 K Now, we have the temperature of the air after the regenerator.

Calculate the turbine work and heat transfer

To calculate the turbine work and heat transfer, we use the equation for turbine efficiency (η_turbine): η_turbine = (W_turbine_out - W_compressor_in) / Q_in Since we know the efficiency of the turbine (90%), we can solve for the heat transfer (Q_in): Q_in = (W_turbine_out - W_compressor_in) / η_turbine We need to determine the work input to the compressor (W_compressor_in) and the work output from the turbine (W_turbine_out). We can use the equations for specific heat at constant pressure (c_p) and variable specific heats for air (c_p2 > c_p1): W_compressor_in = c_p * (T2 - T1) W_turbine_out = c_p * (T3 - T4) Then substitute these terms back into the equation for Q_in and solve for it, using known values of T1 (310 K), T2 (650 K), T3 (1400 K) and T4 (1210 K): Q_in = ((c_p * (T3 - T4)) - (c_p * (T2 - T1))) / η_turbine Q_in = c_p * ( (T3 - T4) - (T2 - T1))/ 0.9

Calculate the heat transfer in the regenerator

Now, we can calculate the amount of heat transfer in the regenerator (Q_regeneration): Q_regeneration = c_p * (T4 - T2) Substitute all given values and solve for Q_regeneration: Q_regeneration = c_p * (1210 K - 650 K) Q_regeneration = c_p * 560 K

Calculate the thermal efficiency of the gas turbine engine

Finally, we can calculate the thermal efficiency (η_thermal) using the following equation: η_thermal = W_net / Q_in First, calculate the net work output (W_net) as the difference between work input to the compressor and work output from the turbine: W_net = W_turbine_out - W_compressor_in Now plug in the previously calculated values and solve for the thermal efficiency: η_thermal = (c_p * ((T3 - T4) - (T2 - T1))) / (c_p * ((T3 - T4) - (T2 - T1)) / 0.9) η_thermal = 0.9 The thermal efficiency of the regenerative gas turbine engine is 90%. To summarize, the heat transfer in the regenerator is c_p * 560 K and the thermal efficiency of the gas turbine engine is 90%.