Question

Using EES (or other) software, determine the effects of pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies on the net work output per unit mass and the thermal efficiency of a simple 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, and draw conclusions from the results.

Short answer

Answer: The variations in pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies significantly influence the performance of the Brayton cycle. A higher pressure ratio and increased maximum cycle temperature generally result in greater net work output per unit mass and improved thermal efficiency. Similarly, higher compressor and turbine efficiencies contribute to better performance, increasing work output and thermal efficiency. However, these parameters need to be optimized for specific applications to achieve the desired results.

Step-by-step solution

Detailed Cycle Analysis

To analyze the Brayton cycle, we need to first determine the individual properties at each stage. These stages are: 1. Inlet of compressor (point 1): \(P_1 = 100 \,\mathrm{kPa}\), \(T_1 = 300\,\mathrm{K}\) 2. Outlet of compressor (point 2) 3. Outlet of combustion chamber (point 3) 4. Outlet of turbine (point 4) The processes in the cycle are: 1-2: Compression in the compressor (adiabatic) 2-3: Heat addition in the combustion chamber (constant pressure) 3-4: Expansion in the turbine (adiabatic) 4-1: Heat rejection in the heat exchanger (constant pressure)

Finding Compressor Outlet Parameters (point 2)

Using the isentropic relations for an ideal compressor and given compressor efficiency, we can determine the temperature and pressure at the outlet of the compressor (point 2). $$T_2 = T_1 * (1+(\gamma-1)/2*(\eta_c)*(P_{r})^\frac{\gamma-1}{\gamma})$$ $$P_2 = P_1 * P_{r}$$ Where: - \(\eta_c\): Compressor efficiency - \(\gamma\): Specific heat ratio of air (value of 1.4) - \(P_{r}\): Pressure ratio

Finding Combustion Chamber Outlet Parameters (point 3)

Since the process is constant pressure in the combustion chamber, we have the pressure: $$P_3 = P_2$$ To find the temperature at the combustion chamber outlet (point 3), we can use the maximum cycle temperature: $$T_3 = T_2 + T_{max}$$ Where: - \(T_{max}\): Maximum cycle temperature

Finding Turbine Outlet Parameters (point 4)

Using the isentropic relations for an ideal turbine and given turbine efficiency, we can determine the temperature and pressure at the outlet of the turbine (point 4). $$P_4 = P_3 / P_{r}$$ $$T_4 = T_3 - \eta_t*(T_3 - T_1)*(1-\frac{1}{P_{r}^{\frac{\gamma-1}{\gamma}}})$$ Where: - \(\eta_t\): Turbine efficiency

Determining Net Work Output Per Unit Mass & Thermal Efficiency

To find the net work output per unit mass (W_net), we first need to find the work of compressor (W_c) and the work of the turbine (W_t): $$W_c = C_p * (T_2 - T_1)$$ $$W_t = C_p * (T_3 - T_4)$$ $$W_{net} = W_t - W_c$$ To find the thermal efficiency, we need to calculate the heat added per unit mass (Q_in) in the combustion chamber: $$Q_{in} = C_p * (T_3 - T_2)$$ Thermal efficiency (\(\eta_{th}\)) can now be calculated: $$\eta_{th} = \frac{W_{net}}{Q_{in}}$$ With the steps described above, we can use software like EES or Matlab to help us determine all possible combinations of pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies, and analyze their effects on the net work output per unit mass and the thermal efficiency of a simple Brayton cycle. Finally, the conclusions can be drawn from the results by studying the dependency of cycle performance on the pressure ratio, maximum temperature, compressor efficiency, and turbine efficiency.

Key concepts

Thermodynamic Efficiency of the Brayton Cycle

The thermodynamic efficiency, or thermal efficiency, of a Brayton cycle is a measure of how effectively it converts the heat added to the cycle into work. It's a vital metric that determines the overall performance and cost-effectiveness of gas turbines used in power plants and jet engines.

Thermal efficiency \( \(\eta_{th}\) \) is calculated using the following equation:\[\eta_{th} = \frac{W_{net}}{Q_{in}}\]Where \(W_{net}\) is the net work output per unit mass and \(Q_{in}\) represents the heat added per unit mass in the combustion chamber. To maximize thermal efficiency, engineers aim to increase the net work output while decreasing the heat input. This is typically done by increasing the temperature at which heat is added or by improving the efficiencies of the compressor and turbine. As these components become more efficient, less work is lost throughout the cycle, increasing the overall thermal efficiency.

Pressure Ratio's Role in Brayton Cycle Performance

The pressure ratio in the Brayton cycle is the ratio of the pressure at the outlet of the compressor to that at the inlet. It has a significant impact on both the net work output and the thermal efficiency of the cycle.

The pressure ratio \( P_{r} \) is defined by the equation:\[P_{r} = \frac{P_2}{P_1}\]An increase in pressure ratio generally leads to a higher thermal efficiency because it increases the temperature at the turbine's inlet, allowing for more work to be extracted from the expanding gases. However, there is a practical limit to the pressure ratio, as exceedingly high values can lead to mechanical and thermal stress on components and potential decreases in component efficiencies. Therefore, identifying the optimal pressure ratio is crucial for the design and operation of an efficient Brayton cycle.

Compressor and Turbine Efficiencies in the Brayton Cycle

The performance of the compressor and turbine is characterized by their efficiencies \(\eta_c\) and \(\eta_t\), respectively. These efficiencies reflect the real-world deviations from the ideal, frictionless processes assumed in theoretical models.

The compressor efficiency \(\eta_c\) affects the temperature at the compressor's outlet, while the turbine efficiency \(\eta_t\) impacts the temperature at the turbine's outlet. High efficiencies indicate that a larger portion of the energy input results in useful work, and thus, plays a pivotal role in enhancing the overall efficiency of the Brayton cycle.Higher efficiencies in these components mean that the cycle can achieve the same degree of pressure rise or drop across the turbine and compressor while consuming less energy in the compressor and extracting more energy in the turbine. This optimization ultimately contributes to a higher thermal efficiency and better cycle performance.

Understanding the Adiabatic Process in the Brayton Cycle

An adiabatic process is a thermodynamic process in which no heat is transferred to or from the working fluid. It is an idealization that provides a foundation for analyzing real-world processes in the Brayton cycle, specifically, the compression in the compressor and the expansion in the turbine.

In an ideal Brayton cycle, both the compression and expansion processes are assumed to be isentropic, which is a reversible adiabatic process. For the compressor:\[T_2 = T_1 * \left(1+(\gamma-1)/2*(\eta_c)*(P_{r})^\frac{\gamma-1}{\gamma}\right)\]and for the turbine:\[T_4 = T_3 - \eta_t*(T_3 - T_1)*\left(1-\frac{1}{P_{r}^{\frac{\gamma-1}{\gamma}}}\right)\]Here, \(\gamma\) is the specific heat ratio of the working fluid (air). In a real Brayton cycle, these processes are not perfectly adiabatic due to friction and other irreversibilities. However, by understanding and improving upon these adiabatic processes, the efficiency and performance of the Brayton cycle can be significantly enhanced.