Question

Consider an ideal gas-turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each stage of the compressor and turbine is 3 The air enters each stage of the compressor at \(300 \mathrm{K}\) and each stage of the turbine at \(1200 \mathrm{K}\). Determine the back work ratio and the thermal efficiency of the cycle, assuming \((a)\) no regenerator is used and \((b)\) a regenerator with 75 percent effectiveness is used. Use variable specific heats.

Short answer

Question: Calculate the back work ratio and thermal efficiency of an ideal gas-turbine cycle with two stages of compression and two stages of expansion, both with and without a regenerator. The pressure ratio across each stage is 3, the temperature at the inlet of each compressor stage is 300K, and the temperature at the inlet of each turbine stage is 1200K. The regenerator has a 75% effectiveness. Answer: To calculate the back work ratio and thermal efficiency of the cycle, both with and without a regenerator, follow these steps: 1. Analyze the Brayton cycle and note down given values. 2. Calculate the temperature ratios for each process. 3. Calculate the work done during each process. 4. Calculate the heat addition. 5. Calculate the back work ratio. 6. Calculate the thermal efficiency of the cycle. Plug in the temperature ratios and specific heat values in Steps 3-6 to get the back work ratio and thermal efficiency of the cycle, both with and without a regenerator.

Step-by-step solution

Analyze the Brayton cycle and note down given values

Given parameters: Pressure ratio across each stage (compressor and turbine): r = 3 Temperature at the inlet of each compressor stage: T1 = T3 = 300K Temperature at the inlet of each turbine stage: T4 = T6 = 1200K

Calculate the temperature ratios for each process

\((a)\) Without a regenerator: Temperature ratios for compressors: \(T_{2}/T_{1}=T_{4}/T_{3}=(r)^{(\gamma-1)/\gamma}\) Temperature ratios for turbines: \(T_{5}/T_{4}=T_{7}/T_{6}=(1/r)^{(\gamma-1)/\gamma}\) \((b)\) With a 75% effective regenerator: Temperature ratios for compressors: \(T_{2}/T_{1}=T'_{4}/T_{3}=(r')^{(\gamma-1)/\gamma}\) Temperature ratios for turbines: \(T_{5}/T_{4'}=T_{7}/T_{6}=(1/r')^{(\gamma-1)/\gamma}\)

Calculate the work done during each process

\((a)\) Without a regenerator: Work done by compressors 1 and 2: \(W_{c1}=c_{p}(T_{2}-T_{1})\), \(W_{c2}=c_{p}(T_{4}-T_{3})\) Work done by turbines 1 and 2: \(W_{t1}=c_{p}(T_{4}-T_{5})\), \(W_{t2}=c_{p}(T_{6}-T_{7})\) \((b)\) With a regenerator: Work done by compressors 1 and 2: \(W_{c1}=c_{p}(T_{2}-T_{1})\), \(W_{c2}=c_{p}(T'_{4}-T_{3})\) Work done by turbines 1 and 2: \(W_{t1}=c_{p}(T_{4}-T_{5})\), \(W_{t2}=c_{p}(T_{6}-T_{7})\)

Calculate the heat addition

\((a)\) Without a regenerator: Heat addition in the combustion chamber: \(Q_{in}=c_{p}(T_{4}-T_{2}+T_{6}-T_{4})\) \((b)\) With a regenerator: Heat addition in the combustion chamber: \(Q_{in}=c_{p}(T_{4}-T'_{4}+T_{6}-T_{4'})\)

Calculate the back work ratio

Back work ratio (BWR) = \( (W_{c1} + W_{c2}) / (W_{t1} + W_{t2})\) \((a)\) Without a regenerator: BWR_no_regenerator = \((W_{c1} + W_{c2}) / (W_{t1} + W_{t2})\) \((b)\) With a regenerator: BWR_with_regenerator = \((W_{c1} + W_{c2}) / (W_{t1} + W_{t2})\)

Calculate the thermal efficiency of the cycle

Thermal efficiency (η) = \(1 - BWR\) \((a)\) Without a regenerator: η_no_regenerator = \(1 - BWR_{no\_regenerator}\) \((b)\) With a regenerator: η_with_regenerator = \(1 - BWR_{with\_regenerator}\) Plug in the temperature ratios and specific heat values in Steps 3-6 to get the back work ratio and thermal efficiency of the cycle, both with and without a regenerator.