Question

Using EES (or other) software, determine the effects of pressure ratio, maximum cycle temperature, regenerator effectiveness, and compressor and turbine efficiencies on the net work output per unit mass and on the thermal efficiency of a regenerative Brayton cycle with air as the working fluid. Air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\) at the compressor inlet. Also, assume constant specific heats for air at room temperature. Determine the net work output and the thermal efficiency for all combinations of the following parameters..

Short answer

, considering constant specific heats at room temperature. This analysis helps us understand how the performance of the cycle is affected by these parameters and provides insights into designing more efficient power cycles.

Step-by-step solution

Understand the Regenerative Brayton Cycle

A Regenerative Brayton Cycle consists of a compressor, a combustion chamber, a turbine, and a regenerator. Air enters the compressor at ambient conditions and gets compressed. It then passes through the regenerator, where it is heated by the exhaust gases before entering the combustion chamber. The high-pressure, high-temperature air enters the turbine, where it expands, doing work. The regenerator effectiveness, compressor efficiency, and turbine efficiency are considered for the cycle analysis.

List the given parameters and constants

1. Pressure at compressor inlet: \(P_1 = 100 \mathrm{kPa}\) 2. Temperature at compressor inlet: \(T_1 = 300 \mathrm{K}\) 3. Constant specific heats for air at room temperature: \(c_p = 1005 \mathrm{J/(kg \cdot K)}\), \(c_v = 717 \mathrm{J/(kg \cdot K)}\), and \(k = 1.4\) (ratio of specific heats) 4. Compressor efficiency: \(\eta_c\) 5. Turbine efficiency: \(\eta_t\) 6. Regenerator effectiveness: \(\epsilon\) 7. Maximum cycle temperature: \(T_{max}\) 8. Pressure ratio: \(PR\)

Calculating the output and efficiency

For each combination of pressure ratio, maximum cycle temperature, regenerator effectiveness, compressor efficiency, and turbine efficiency, we need to perform the following calculations: 1. Compute the temperature after the compressor (\(T_2\)) using the formula: \(T_2 = T_1 \cdot \left(PR^\frac{k-1}{k}\right)^{\frac{\eta_c}{1-\eta_c}}\) 2. Compute the temperature after the regenerator (\(T_3\)) using the formula: \(T_3 = T_1 + \epsilon \cdot (T_2 - T_1)\) 3. Compute the temperature after the combustion chamber (\(T_4\)): \(T_4 = T_{max}\) 4. Compute the temperature after the turbine (\(T_5\)) using the formula: \(T_5 = T_4 - \eta_t \cdot (T_3 - T_1)\) 5. Compute the net work output per unit mass (\(W_{net}\)) using the formula: \(W_{net} = c_p \cdot (T_3 - T_1) - c_p \cdot (T_2 - T_1) \) 6. Compute the heat input per unit mass (\(Q_{in}\)) using the formula: \(Q_{in} = c_p \cdot (T_4 - T_3)\) 7. Compute the thermal efficiency (\(\eta_{th}\)) using the formula: \(\eta_{th} = \frac{W_{net}}{Q_{in}}\) Repeat these calculations for all combinations of the given parameters and record the results. In conclusion, by performing these calculations for all possible combinations of the given parameters, we can determine the net work output per unit mass and the thermal efficiency of a regenerative Brayton cycle with air as the working fluid

Key concepts

Thermal Efficiency

When we discuss thermal efficiency in the context of a regenerative Brayton cycle, we refer to the measure of how effectively a heat engine converts the thermal energy input into work output. This efficiency is of paramount importance as it directly impacts the performance and cost-effectiveness of power plants and engines.

Using the thermal efficiency formula \( \eta_{th} = \frac{W_{net}}{Q_{in}} \), we relate the net work output per unit mass to the input heat per unit mass. The regenerator plays a significant role here, as it recovers energy from the exhaust and preheats the air entering the combustion chamber. A higher regenerator effectiveness means more recovery of usable energy, thus improving the thermal efficiency of the cycle. Compressor and turbine efficiencies also contribute directly to the cycle's thermal performance—higher efficiencies mean less energy lost to irreversibilities and mechanical losses, boosting the overall energy conversion rate.

Net Work Output

Net work output is a crucial performance indicator for any power cycle, reflecting the usable energy that can be harnessed to perform external work after accounting for all losses within the system. In a regenerative Brayton cycle, the net work output per unit mass is given by the equation \( W_{net} = c_p \cdot (T_3 - T_1) - c_p \cdot (T_2 - T_1) \).

It is the difference in the work produced by the turbine and the work consumed by the compressor. By optimizing parameters like pressure ratio, maximum cycle temperature, and regenerator effectiveness, along with improving compressor and turbine efficiencies, the net work output can be maximized. For example, a higher pressure ratio generally increases net work output, but it also demands more from the compressor. The exercise aims to find a balance that yields the best performance for the specific conditions.

Compressor and Turbine Efficiencies

In the regenerative Brayton cycle, the compressor and turbine efficiencies are critical factors in dictating the overall performance of the cycle. Efficiency for these components is defined as the ratio of the actual work to the ideal work—if they were isentropic, meaning no entropy would be produced during compression or expansion.

The formulae incorporating these efficiencies, \( T_2 = T_1 \cdot \left(PR^{\frac{k-1}{k}}\right)^{\frac{\eta_c}{1-\eta_c}} \) for the compressor and \( T_5 = T_4 - \eta_t \cdot (T_3 - T_1) \) for the turbine, show how they affect temperatures and subsequently the net work output. Generally, as these efficiencies increase so does the overall cycle efficiency, because less thermal energy is wasted. Therefore, enhancing compressor and turbine designs to achieve higher efficiencies is an important aspect of improving the regenerative Brayton cycle's performance.

Regenerator Effectiveness

Regenerator effectiveness is defined as the ability of the regenerator to transfer heat from the turbine's exhaust to the compressed air before it enters the combustion chamber. It effectively measures how well the regenerator is able to approach the ideal scenario where the exit temperature of the gas from the regenerator equals the entrance temperature of the gas into the turbine.

Expressed by the formula \( T_3 = T_1 + \epsilon \cdot (T_2 - T_1) \) , regenerator effectiveness plays a significant role in cycle efficiency. The closer its value is to 1 (or 100%), the better the recovery of heat from the exhaust gases. This reduces the fuel requirements and consequently increases thermal efficiency. However, practical considerations, such as cost and complexity, may limit how effective the regenerator can be made. Therefore, it's crucial to optimize regenerator design within these constraints to maximize the benefits for the regenerative Brayton cycle's operation.