Question

An ideal gas turbine cycle with many stages of compression and expansion and a regenerator of 100 percent effectiveness has an overall pressure ratio of \(10 .\) Air enters every stage of compressor at \(290 \mathrm{K}\), and every stage of turbine at \(1200 \mathrm{K}\). The thermal efficiency of this gas-turbine cycle is \((a) 36\) percent (b) 40 percent \((c) 52\) percent \((d) 64\) percent \((e) 76\) percent

Short answer

Answer: The thermal efficiency is approximately 64%.

Step-by-step solution

Calculate the work input in each stage of the compressor

Using the adiabatic relation for the ideal gas, we can calculate the work input in each stage of the compressor. The relationship between temperature and pressure for an adiabatic process in an ideal gas can be represented as: $$ T_2= T_1\left(\frac{P_2}{P_1}\right)^{(k-1)/k} $$ Where \(T_1\) is the initial temperature (290 K), \(P_1\) is the initial pressure, \(T_2\) is the final temperature after compression, \(P_2\) is the final pressure, and \(k\) is the specific heat ratio for air which is approximately 1.4 for an ideal gas. We are given the overall pressure ratio (\(P_2/P_1\)) between the compressor and the turbine as 10. So, we can rewrite this equation as: $$ T_2= T_1\left(\frac{P_2}{P_1}\right)^{(k-1)/k} $$ $$ T_2= 290\left(\frac{10}{1}\right)^{(1.4-1)/1.4} $$ $$ T_2\approx 580.54 \mathrm{K}\ $$ Now we can determine the work input in each stage of the compressor as: $$ W_{in} = (k-1) \times P_1 \times (T_2 - T_1) $$

Calculate the work output in each stage of the turbine

Using a similar approach, we can calculate the work output in each stage of the turbine. The relationship between temperature and pressure for an adiabatic process in an ideal gas turbine cycle is: $$ T_4= T_3\left(\frac{P_4}{P_3}\right)^{(k-1)/k} $$ Where \(T_3\) is the initial temperature (1200 K), \(P_3\) is the initial pressure, \(T_4\) is the final temperature after expansion, \(P_4\) is the final pressure. Using the given overall pressure ratio, we can find \(T_4\): $$ T_4= T_3\left(\frac{P_4}{P_3}\right)^{(k-1)/k} $$ $$ T_4= 1200\left(\frac{1}{10}\right)^{(1.4-1)/1.4} $$ $$ T_4 \approx 669.46 \mathrm{K}\ $$ Now we can determine the work output in each stage of the turbine as: $$ W_{out} = (k-1) \times P_3 \times (T_3 - T_4) $$

Calculate the thermal efficiency

Thermal efficiency (\(\eta\)) is the ratio of the net work output to the heat input (Q_in) in the cycle. $$ \eta = \frac{W_{out} - W_{in}}{Q_{in}} $$ For an ideal gas with a regenerator of 100% effectiveness, the heat input (Q_in) can be determined as: $$ Q_{in} = C_p \times (T_3 - T_2) $$ Where \(C_p\) is the specific heat capacity at constant pressure for air which is approximately \(1.005 \mathrm{kJ/(kg\times K)}\). So, $$ \eta = \frac{(k-1) \times P_3 \times (T_3 - T_4) - (k-1) \times P_1 \times (T_2 - T_1)}{C_p \times (T_3 - T_2)} $$ Simplifying and inserting the known values, we get the thermal efficiency: $$ \eta \approx \frac{1.005 \times (1200 - 580.54)}{(1.4-1)(1200-669.46) - (1.4-1)(580.54 - 290)} $$ $$ \eta \approx 0.6396 $$ So the thermal efficiency is approximately \(64\%\). Hence, the correct option is: \((d) 64\) percent