Question

In an ideal Brayton cycle with regeneration, argon gas is compressed from \(100 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\) to \(400 \mathrm{kPa}\), and then heated to \(1200^{\circ} \mathrm{C}\) before entering the turbine. The highest temperature that argon can be heated in the regenerator is (a) \(246^{\circ} \mathrm{C}\) (b) \(846^{\circ} \mathrm{C}\) \((c) 689^{\circ} \mathrm{C}\) \((d) 368^{\circ} \mathrm{C} \quad(e) 573^{\circ} \mathrm{C}\)

Short answer

Answer: The highest temperature that argon can be heated in the regenerator is approximately 831.85°C.

Step-by-step solution

Calculate the Temperature and Pressure after Compression

To find the temperature after compression, we need to use the isentropic relation between the pressure, temperature, and the specific heat ratio, assuming the compression process is isentropic. The specific heat ratio of argon is \(\gamma = 1.67\), and the initial pressure and temperature are given as \(P_1 = 100\,\mathrm{kPa}\) and \(T_1 = 25^\circ\mathrm{C}\). The final pressure is given as \(P_2 = 400\,\mathrm{kPa}\), we can use the isentropic relation to find \(T_2\): $$ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma - 1}{\gamma}} $$ Now, we can plug in the values and calculate \(T_2\): $$ \frac{T_2}{25 + 273.15} = \left(\frac{400}{100}\right)^{\frac{1.67 - 1}{1.67}} $$ $$ T_2 = 298.15 \cdot \left(4\right)^{\frac{0.67}{1.67}} = 368\,\mathrm{K} $$ Note that we need to convert temperatures from Celsius to Kelvin for these calculations. So, \(T_2 = 368\,\mathrm{K} = 95^\circ\mathrm{C}\), which is the temperature of argon at the end of the compression process.

Calculate the Highest Temperature in the Regenerator

Now, we can use the temperature of the argon at the end of the compression process (\(T_2 = 95^\circ\mathrm{C}\)) and the temperature before entering the turbine (\(T_3 = 1200^\circ\mathrm{C}\)) to determine the highest temperature that argon can be heated in the regenerator. The regenerator is essentially a counter-flow heat exchanger; thus, in the best-case scenario, the temperature of the compressed argon can approach the temperature of the argon before entering the turbine, giving the maximum possible effectiveness for the regenerator. The highest temperature that argon can be heated in the regenerator (\(T_{max}\)) can now be calculated: $$ T_{max} = T_3 - T_2 $$ $$ T_{max} = 1200 - 95 = 1105\,\mathrm{K} $$ Now, we can convert the temperature back to Celsius: $$ T_{max} = 1105 - 273.15 = 831.85^\circ\mathrm{C} $$ Thus, the highest temperature that argon can be heated in the regenerator is around \(831.85^\circ\mathrm{C}\). Looking at the available options, the closest answer is option (b) \(846^\circ\mathrm{C}\).