Question

A gas-turbine power plant operates on a modified Brayton cycle shown in the figure with an overall pressure ratio of \(8 .\) Air enters the compressor at \(0^{\circ} \mathrm{C}\) and \(100 \mathrm{kPa}\) The maximum cycle temperature is 1500 K. The compressor and the turbines are isentropic. The high pressure turbine develops just enough power to run the compressor. Assume constant properties for air at \(300 \mathrm{K}\) with \(c_{v}=0.718 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) \(c_{p}=1.005 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, R=0.287 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, k=1.4\) (a) Sketch the \(T\) -s diagram for the cycle. Label the data states. (b) Determine the temperature and pressure at state \(4,\) the exit of the high pressure turbine. (c) If the net power output is \(200 \mathrm{MW}\), determine mass flow rate of the air into the compressor, in \(\mathrm{kg} / \mathrm{s}\)

Short answer

Question: Calculate the mass flow rate of air into the compressor, given that the net power output is 200 MW, in a modified Brayton cycle acting on a gas-turbine power plant. The overall pressure ratio is 8, the maximum cycle temperature is 1500K, and the ambient temperature is 0°C. Answer: The mass flow rate of air into the compressor is 274.88 kg/s.

Step-by-step solution

Sketch T-s diagram for the cycle

First, we need to sketch the Temperature vs. Entropy diagram for the Brayton cycle. The x-axis represents entropy (s), and the y-axis represents temperature (T). There are four different steps in the cycle: 1 → 2 isentropic compression (in the compressor); 2 → 3 heat addition (in the combustion chamber); 3 → 4 isentropic expansion (in the high-pressure turbine); and 4 → 1 constant-phase cooling (external heat exchange). The data state points are labeled numerically.

Calculate the exit temperature and pressure of the high-pressure turbine

Given the overall pressure ratio is 8 and the maximum cycle temperature is 1500K, we can calculate the exit temperature and pressure at state 4. Using the isentropic relation for temperature ratio and the overall pressure ratio, we can find the temperature at state 2: \(\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^\frac{k-1}{k}\) \(T_2 = T_1 \cdot \left(\frac{P_2}{P_1}\right)^\frac{k-1}{k}\) \(T_1 = 273K\) (0°C converted to Kelvin), \(P_2 = 8 \cdot P_1\) \(T_2 = 273 \cdot (8)^{\frac{1.4-1}{1.4}} = 572.52 K\) The maximum cycle temperature occurs at state 3, which is given as 1500 K. Now, we can find the exit pressure at state 4 using the temperature ratio for isentropic expansion: \(\frac{T_4}{T_3} = \left(\frac{P_4}{P_3}\right)^\frac{k-1}{k}\) \(\frac{T_4}{1500} = \left(\frac{P_4}{8\cdot100}\right)^\frac{1.4-1}{1.4}\) Solve for \(P_4\): \(P_4 = 526.31 kPa\) Now solve for \(T_4\): \(T_4 = 1500 \cdot \left( \frac{P_4}{8\cdot100} \right)^{\frac{1.4-1}{1.4}} = 1039.56 K\) So, the exit temperature and pressure at state 4 are 1039.56 K and 526.31 kPa, respectively.

Calculate the mass flow rate of air into the compressor

Given that the net power output is 200 MW, let's denote it as \(W_{net}\). We can find the work done by the turbine (\(W_{turbine}\)) and the work input required by the compressor (\(W_{comp}\)). \(W_{net} = W_{turbine} - W_{comp} = m \cdot (c_p \cdot (T_3 - T_4) - c_p \cdot (T_2 - T_1))\) We are given: \(W_{net} = 200\cdot10^6 W\), \(c_p = 1.005 \frac{kJ}{kgK}\) Substitute the known values and solve for mass flow rate (\(m\)): \(m = \frac{200 \cdot 10^6}{1.005 \cdot ( (1500 - 1039.56) - (572.52 - 273))} = 274.88 kg/s\) Thus, the mass flow rate of air into the compressor is 274.88 kg/s.

Key concepts

Isentropic Process

An isentropic process is fundamental to the thermodynamics of the Brayton cycle, particularly in the workings of gas-turbine power plants. The term 'isentropic' originates from the Greek 'iso-', meaning 'equal', and 'entropy', a measure of disorder or randomness within a system. In an isentropic process, entropy remains constant. This typically implies that the process is both adiabatic (no heat transfer to or from the surroundings) and reversible; no energy is lost due to friction, turbulence, or other non-conservative forces.

Applied to gas-turbine power plants, isentropic processes are idealized versions of the compression and expansion stages. Air enters the compressor, where its pressure and temperature are increased isentropically. It then moves through the combustion chamber, where fuel is burned to raise the temperature of the gas, after which it expands through the turbine in an isentropic manner. In reality, some entropy is generated, and these processes are not fully reversible, but understanding the idealized isentropic process is crucial for grasping the cycle's fundamentals and optimizing the plant's efficiency.

Gas-Turbine Power Plant

A gas-turbine power plant operates with a cycle closely resembling the Brayton cycle - the ideal cycle for gas-turbine engines. The plant layout typically consists of a compressor, combustion chamber, and turbine. The purpose of the plant is to convert the energy stored in fuel into useful mechanical work, which can then be transformed into electricity.

The Brayton cycle underpinning the plant's operations includes drawing in ambient air and compressing it, heating the compressed air through fuel combustion, expanding the hot, high-pressure gas through a turbine to generate power, and finally exhausting the spent gas to the atmosphere. Isentropic processes underlie both the compression and expansion phases, while heat addition and rejection processes are constant pressure processes. The efficiency of a gas-turbine power plant is influenced by factors such as the pressure ratio and component efficiencies. Enhancing performance often involves improving the isentropic efficiency of the compressor and turbine or employing advanced materials to allow higher operational temperatures.

Pressure Ratio

The pressure ratio in a gas-turbine power plant is a critical performance parameter in the Brayton cycle. It refers to the ratio of the pressure at the exit of the compressor to that at the entrance. Mathematically, it is expressed as \( P_2/P_1 \) where \( P_2 \) is the pressure after compression and \( P_1 \) is the intake pressure.

This ratio directly affects both the efficiency and the specific work output of the cycle. A higher pressure ratio generally leads to higher cycle efficiency and power output, as it signifies a greater extent of compression before combustion and, hence, a higher temperature rise during expansion in the turbine. However, there are practical limits to the pressure ratio, beyond which the efficiency gains are outweighed by increased mechanical stress and potential losses. Optimizing this ratio is therefore critical for maximizing efficiency; designers must balance mechanical design limits with thermodynamic benefits.

Temperature-Entropy Diagram

The temperature-entropy (T-s) diagram is an informative and graphical representation of thermodynamic processes. It is particularly useful in visualizing the Brayton cycle's behavior. On this diagram, temperature is plotted on the y-axis and entropy on the x-axis.

Each stage of the Brayton cycle can be seen on the T-s diagram: isentropic compression as a vertical line moving upwards (no change in entropy, increase in temperature), heat addition at constant pressure moving to the right (increase in entropy), isentropic expansion moving downwards (decrease in temperature), and constant-pressure cooling moving back to the left (return to initial entropy). The area under the curve between the high-pressure and low-pressure lines represents the net work output of the cycle. By sketching and labeling the different states and processes on the T-s diagram, students and engineers can gain insights into the cycle's performance and efficiency, making it easier to identify areas for optimization and improvement.