Question

Consider a gas turbine that has a pressure ratio of 6 and operates on the Brayton cycle with regeneration between the temperature limits of 20 and \(900^{\circ} \mathrm{C}\). If the specific heat ratio of the working fluid is \(1.3,\) the highest thermal efficiency this gas turbine can have is \((a) 38\) percent (b) 46 percent \((c) 62\) percent \((d) 58\) percent \((e) 97\) percent

Short answer

Answer: 62 percent.

Step-by-step solution

Recall the Brayton Cycle

The Brayton cycle consists of four processes: (1) adiabatic compression, (2) constant pressure heat addition, (3) adiabatic expansion, and (4) constant pressure heat rejection.

Calculate the temperature ratio for adiabatic processes

For adiabatic processes, the relationship between temperature and pressure ratio is given by the following equation: \(T_2 / T_1 = (P_2 / P_1)^{(k-1)/k}\) where \(T_1\) and \(T_2\) are the initial and final temperatures, \(P_1\) and \(P_2\) are the initial and final pressures, \(k\) is the specific heat ratio. From the given pressure ratio (\(P_2 / P_1 = 6\)) and specific heat ratio (\(k = 1.3\)), we can calculate the temperature ratio: \(T_2 / T_1 = (6)^{(1.3-1)/1.3}\)

Calculate the final temperatures after adiabatic processes

We are given the initial temperature \(T_1 = 20 + 273.15 = 293.15 K\). Using the temperature ratio, we calculate the final temperature after adiabatic compression: \(T_2 = T_1 * (6)^{(1.3-1)/1.3} = 293.15 * (6)^{(0.3)/1.3} \approx 481.26 K\) The maximum temperature of the Brayton cycle is given as \(900^{\circ} \mathrm{C} = 1173.15 K\). Using the temperature ratio, we find the final temperature after adiabatic expansion: \(T_3 / T_4 = (6)^{(1.3-1)/1.3}\) \(T_4 = T_3 / (6)^{(1.3-1)/1.3} = 1173.15 / (6)^{(0.3)/1.3} \approx 712.90 K\)

Calculate the work done and heat input during the cycle

The net work done during the cycle consists of the work done during adiabatic compression and adiabatic expansion: \(W_{net} = mC_p(T_3 - T_4) - mC_v(T_2 - T_1)\) where \(C_p\) and \(C_v\) are the specific heats at constant pressure and volume, and \(m\) is the mass of the working fluid. Given a specific heat ratio (\(k = 1.3\)), the relationship between \(C_p\) and \(C_v\) is \(C_p / C_v = k\). So, \(C_p = k * C_v\). Now, we can rewrite the net work done equation as: \(W_{net} = mC_v[(k)(T_3 - T_4) - (T_2 - T_1)]\) The heat input to the cycle occurs during the constant pressure heat addition process: \(Q_{in} = mC_p(T_3 - T_2)\) Substituting \(C_p = k * C_v\), we have: \(Q_{in} = mkC_v(T_3 - T_2)\)

Calculate the highest thermal efficiency of the gas turbine

Thermal efficiency is the ratio of the net work done during the cycle to the heat input: \(\eta_{th} = W_{net} / Q_{in}\) Substituting the \(W_{net}\) and \(Q_{in}\) expressions from steps 4, we have: \(\eta_{th} = \frac{(k)(T_3 - T_4) - (T_2 - T_1)}{k(T_3 - T_2)}\) Using the temperatures found in step 3: \(\eta_{th} = \frac{(1.3)(1173.15 - 712.90) - (481.26 - 293.15)}{(1.3)(1173.15 - 481.26)} \approx 0.62\) The highest thermal efficiency of the gas turbine is approximately 62%. The correct answer is (c) 62 percent.

Key concepts

Adiabatic Processes

Adiabatic processes are fundamental to understanding the Brayton cycle efficiency. An adiabatic process is one in which no heat is exchanged with the surroundings; hence, all the work done on or by the system affects its internal energy directly. In the Brayton cycle, both the compression and the expansion processes are assumed to be adiabatic. During compression, work is done on the gas, increasing its temperature without the addition of external heat. Conversely, during expansion, the gas performs work by expelling energy, leading to a temperature drop without heat loss.

The relationship between pressure and temperature in an adiabatic process for an ideal gas is expressed by \(T_2 / T_1 = (P_2 / P_1)^{(k-1)/k}\), where \(k\) is the specific heat ratio. This ratio indicates the change in temperature as the gas is compressed or expanded without heat transfer. This concept is pivotal in calculating the temperature changes that occur in the Brayton cycle's adiabatic stages, which are key to determining the cycle's efficiency.

Thermal Efficiency

Thermal efficiency is a dimensionless performance metric that indicates how well a thermodynamic cycle converts heat into work. It's defined as the ratio of net work output over the heat input. In the context of the Brayton cycle, the thermal efficiency represents the efficiency of a gas turbine in converting the energy from the fuel into mechanical energy.

The theoretical thermal efficiency of the Brayton cycle can be improved by increasing the maximum temperature of the cycle or by increasing the pressure ratio of the compressor and turbine. However, the efficiency also depends on the specific heat ratio of the working fluid and the effectiveness of the regenerator. In the problem at hand, calculating the Brayton cycle's thermal efficiency involves determining the temperature changes during adiabatic processes and the heat added during combustion. By using the formula \(\eta_{th} = W_{net} / Q_{in}\), we directly relate the work and heat transfer within the cycle to measure its efficiency.

Specific Heat Ratio

The specific heat ratio, denoted by \(k\), is a material-specific value that compares the heat capacity at constant pressure \(C_p\) to the heat capacity at constant volume \(C_v\). For an ideal gas, \(k=C_p/C_v\), and its value typically ranges between 1.3 and 1.67 for diatomic gases, which are commonly used in Brayton cycle applications. The specific heat ratio is crucial because it affects the magnitude of temperature changes during adiabatic processes. Gases with higher values of \(k\) will experience greater temperature changes during compression and expansion for a given pressure change.

In the exercise, we are provided with a specific heat ratio of 1.3, which directly influences the calculation of temperatures after adiabatic processes and, subsequently, the thermal efficiency of the cycle. It is important to note that the specific heat ratio is not merely a factor in equations; it represents the physical properties of the working fluid and thus significantly impacts the performance of real-world thermodynamic cycles.