Question

A gas turbine operates with a regenerator and two stages of reheating and intercooling. This system is designed so that when air enters the compressor at \(100 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\) the pressure ratio for each stage of compression is \(3 ;\) the air temperature when entering a turbine is \(500^{\circ} \mathrm{C} ;\) and the regenerator operates perfectly. At full load, this engine produces \(800 \mathrm{kW} .\) For this engine to service a partial load, the heat addition in both combustion chambers is reduced. Develop an optimal schedule of heat addition to the combustion chambers for partial loads ranging from 400 to \(800 \mathrm{kW}\)

Short answer

First, analyze the given conditions of the gas turbine and then calculate the temperature and pressures for different stages under full load conditions. After finding the temperatures at different stages for full load, calculate the optimal heat addition schedule for partial loads by proportionally decreasing the amount of heat added in both combustion chambers as the load decreases from 800 kW to 400 kW. Keep the proportion of heat addition in each combustion chamber constant when adjusting the required heat addition for optimal engine operation.

Step-by-step solution

Analyze the given conditions of the gas turbine

When solving the problem, we need to take into account the following: - Air enters the compressor at 100 kPa and 15°C - Pressure ratio for each stage of compression is 3 - Air temperature when entering a turbine is 500°C - Perfect regenerator - Full load output is 800 kW

Calculate the temperature and pressures for different stages under full load conditions

To do this, we will first calculate the pressure in different stages using the pressure ratios given: - After 1st stage of compression: \(P_1 = 100 \times 3 = 300 \, \mathrm{kPa}\) - After 2nd stage of compression: \(P_2 = 300 \times 3 = 900 \, \mathrm{kPa}\) Next, we need to calculate the temperature at different stages. Since the air temperature entering a turbine is given, we have: - Air temperature after 2nd stage of reheating: \(T_2 = 500 \, ^{\circ}C\) The regenerator operates perfectly; therefore, we can assume that the temperature after the 1st stage intercooler is the same as the temperature at the beginning: - Air temperature after 1st stage of intercooler: \(T_1' = 15 \, ^{\circ}C\) Now, we can calculate the temperatures after each stage of compressor and reheater using these temperatures. We need to use isentropic relations since we are given pressure ratios: $$T_1 = T_1'\times \left(\frac{P_1}{100}\right)^{(\gamma-1)/\gamma} = 15 \times \left(\frac{300}{100}\right)^{(\gamma-1)/\gamma}$$ $$T_2' = T_1\times (\frac{P_2}{P_1})^{(\gamma-1)/\gamma} = T_1 \times \left(\frac{900}{300}\right)^{(\gamma-1)/\gamma}$$ $$T_2 = T_1'\times \left(\frac{P_2}{100}\right)^{(\gamma-1)/\gamma} = 15 \times \left(\frac{900}{100}\right)^{(\gamma-1)/\gamma}$$ Here, \(\gamma\) is the specific heat ratio for air, which is typically around 1.4. Calculate the values of \(T_1\), \(T_2'\), and \(T_2\) using the given formula.

Calculate the optimal schedule of heat addition for partial loads between 400 kW and 800 kW

Now that we have the temperatures at different stages for full load, we need to find the optimal heat addition schedule for partial loads. As the load decreases from 800 kW to 400 kW, we need to proportionally decrease the amount of heat added in both combustion chambers. Let the heat addition in each combustion chamber be \(Q_1\) and \(Q_2\) at full load condition (800 kW). For any partial load \(P_{load}\) (between 400 kW and 800 kW), we can calculate the required heat addition in each combustion chamber: $$Q_{1p} = Q_1 \times \frac{P_{load}}{800}$$ $$Q_{2p} = Q_2 \times \frac{P_{load}}{800}$$ Remember to keep the proportion of heat addition in each combustion chamber constant when adjusting \(Q_{1p}\) and \(Q_{2p}\), as this allows for optimal engine operation. Calculate \(Q_{1p}\) and \(Q_{2p}\) based on the given partial load. This will provide the optimal schedule of heat addition in the combustion chambers. By following these steps, you can find the optimal schedule of heat addition in the combustion chambers for gas turbines with regenerators and two stages of reheating and intercooling under different loads between 400 kW and 800 kW.

Key concepts

Regenerator in Gas Turbines

In a gas turbine, a regenerator plays a crucial role in enhancing efficiency by recycling heat from the exhaust gases to preheat the compressed air before it enters the combustion chamber. This clever heat exchange process means that less fuel is needed to reach the desired turbine inlet temperature, as the pre-warmed air already holds some of the energy that would otherwise come from the fuel alone.

The perfect regenerator, as described in the exercise, is an idealized component where the heat transfer from the exhaust to the compressed air is 100% efficient, with no losses. Although this isn't achievable in practice, high-efficiency regenerators can significantly improve a gas turbine's thermal efficiency, making them valuable in real-world applications.

Reheating and Intercooling

Reheating and intercooling are techniques used in gas turbines to further improve efficiency. Reheating refers to the process of heating the working fluid (air) again after it has expanded through part of the turbine. This reheated air then enters another turbine stage, extracting more work and increasing the overall output of the system.

Intercooling, on the other hand, is applied between compressor stages. It cools the air that has been compressed and heated up in the first stage, reducing the work required for further compression in the second stage. The cooled air, having a higher density, allows more mass flow through the second stage for a given volume, effectively improving the compressor's performance. Together, reheating and intercooling make the gas turbine more efficient, especially under full load conditions described in the exercise.

Partial Load Operation

Gas turbines may not always run at their full capacity, often operating under partial load conditions. The partial load operation must be managed carefully to maintain efficiency and avoid operational issues. The exercise outlines a scenario where a gas turbine reduces the heat addition to both combustion chambers as a way to handle partial loads ranging from 400 kW to 800 kW.

The challenge here is to develop an optimal schedule of heat addition that ensures the engine still runs effectively at these lower loads. By proportionally adjusting the fuel supply to match the power demand, engineers can optimize the turbine's performance and maintain its reliability. This approach prevents excessive wear on the turbine components and allows for efficient operation across a range of power outputs.

Isentropic Relations

Isentropic processes are reversible and adiabatic, meaning there is no heat transfer into or out of the system, and the total entropy remains constant. In the context of gas turbines, applying isentropic relations aids in predicting the temperature changes across compressors and turbines under ideal conditions.

In the step by step solution, we see the use of isentropic relations to calculate the temperatures after each compression stage, given the pressure ratios and assuming adiabatic conditions. These relations depend on the specific heat ratio, \( \gamma \), and the exercise carefully employs these formulas to determine the temperatures that are crucial for outlining the engine's performance at different loads.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts the heat from fuel into work. For gas turbines, thermal efficiency is pivotal as it represents the effectiveness of the engine in transforming fuel's thermal energy into mechanical energy. High thermal efficiency in turbines translates to lower fuel costs and reduced environmental impact due to decreased CO2 emissions per unit of power generated.

Enhancements like the regenerator, reheating, and intercooling all serve to boost the overall thermal efficiency of gas turbines. By preheating the air, reintroducing heat, and cooling the air between compression stages, these systems extract more work from a given amount of fuel. In essence, they all work towards the goal of improving the thermal efficiency of the gas turbine, which is a primary concern in both design and operation.

Combustion Chamber Heat Addition

The combustion chamber is where the fuel mixes with the high-pressure air and is ignited to produce the high-temperature gases needed to drive the turbine blades. In gas turbines, controlling the heat addition in the combustion chamber is vital for achieving optimal engine performance across varying loads.

As seen in the exercise, the amount of heat added, represented by \( Q_1 \) and \( Q_2 \) at full load, must be adjusted when the engine is running at partial loads. This careful modulation ensures the temperature of the gases entering the turbine stages remains within proper limits, maintaining the engine's efficiency and performance. Determining the appropriate reduction in heat addition, proportional to the decreased load, is essential for maintaining engine efficiency and longevity during partial load operations.