Question

Compare the thermal efficiency of a two-stage gas turbine with regeneration, reheating and intercooling to that of a three-stage gas turbine with the same equipment when \((a)\) all components operate ideally, \((b)\) air enters the first compressor at \(100 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C},(c)\) the total pressure ratio across all stages of compression is \(16,\) and \((d)\) the maximum cycle temperature is \(800^{\circ} \mathrm{C}\)

Short answer

Question: Compare the thermal efficiency of a two-stage gas turbine and a three-stage gas turbine with regeneration, reheating, and intercooling, considering the given conditions. Answer: Based on the calculations, the thermal efficiency for both the two-stage and three-stage gas turbines is approximately 0.727, assuming ideal conditions. Therefore, both systems have the same thermal efficiency under the given conditions.

Step-by-step solution

Calculate the thermal efficiency of a two-stage gas turbine

Given that all components operate ideally, we can use the following formula to calculate the thermal efficiency for an ideal gas turbine: $$\eta_{thermal} = 1 - \frac{T_{1}}{T_{max}}$$ where T₁ is the initial temperature and Tₘₐₓ is the maximum cycle temperature. The initial temperature is given: 20°C + 273.15 = 293.15 K. The maximum cycle temperature is given: 800°C + 273.15 = 1073.15 K. Now, we can calculate the thermal efficiency of an ideal two-stage gas turbine: $$\eta_{2-stage} = 1 - \frac{293.15}{1073.15} \approx 0.727$$

Calculate the thermal efficiency of a three-stage gas turbine

In the problem, we are given the conditions for an ideal three-stage gas turbine. However, to calculate its thermal efficiency, we'll first find the pressure ratio across each stage. For a three-stage turbine, the total pressure ratio is \(r_t = r_1 \cdot r_2 \cdot r_3\). We know that \(r_t = 16\). Assuming equal pressure ratio for each stage, we can find the pressure ratio for each stage: $$r_1 = r_2 = r_3 = \sqrt[3]{16} \approx 2.52$$ Since all components operate ideally, the thermal efficiency formula is the same as for the two-stage system: $$\eta_{3-stage} = 1 - \frac{T_{1}}{T_{max}} \approx 0.727$$

Compare the thermal efficiency

As per the calculations, the thermal efficiency of the two-stage gas turbine and the three-stage gas turbine with regeneration, reheating and intercooling is the same, approximately equal to 0.727. Thus, under the given ideal conditions, both systems have the same thermal efficiency.

Key concepts

Regeneration Gas Turbine

In the quest for improved efficiency, regeneration in gas turbines plays a vital role. A regeneration gas turbine system recovers some of the waste heat from the exhaust to preheat the air entering the combustion chamber. This process ensures that less fuel is required to reach the combustion temperature, leading to better fuel economy and higher thermal efficiency.

Imagine you're heating up a pot of soup; rather than starting from scratch with a cold pot each time, you keep the pot warm using some of the heat from the last batch. That's essentially what regeneration does – it recycles heat, reducing the energy needed to heat the air to the desired temperature. By implementing regeneration, we can approach the thermal efficiencies calculated in Step 1, which often makes this approach attractive in real-world applications.

Reheating Gas Turbine

On the other hand, reheating in gas turbines involves reintroducing heat between turbine stages. Picture a relay race where after the first runner, instead of losing momentum, you give the baton (in this case, the working fluid) a fresh burst of energy before handing it off to the next runner.

This method is used to increase the power output without increasing the peak cycle temperature. Reheating helps the turbine achieve greater work output by taking advantage of the increased volume flow at later turbine stages. While it does not directly impact the thermal efficiencies we discuss in the Ideal cycle for this exercise, it's essential to note that the real-world applications could see a boost in performance.

Intercooling Gas Turbine

Intercooling is yet another technique aimed at enhancing gas turbine efficiency. It involves cooling the air between compressor stages. When air is compressed, its temperature rises, and if we cool it down before compressing it further, we can reduce the amount of work the subsequent compressor stages need to do.

Think of taking a break to cool off in the shade after a sprint before running again; you'll likely perform better than if you had just kept running without a break. By incorporating intercooling, the potential for improved efficiency surfaces, but as seen in the solution given, under ideal conditions, it doesn't alter the theoretical thermal efficiency ratio for the configurations compared.

Pressure Ratio

The pressure ratio is a critical parameter in a gas turbine operation. Technically, it represents the ratio of the pressure at the turbine's outlet to the pressure at the inlet. A higher pressure ratio usually leads to higher thermal efficiency because it allows the turbine to extract more work from the high-pressure gas.

However, this comes with a trade-off. Increasing the pressure ratio also increases the temperature of the gas, which can lead to material challenges and reduced component life due to thermal stress. As we calculated in Step 2, assuming equal pressure ratios across each stage simplifies the analysis, but the real-world application requires a delicate balancing act to optimize this parameter while considering material and design constraints.

Maximum Cycle Temperature

The maximum cycle temperature is the highest temperature achieved within the thermodynamic cycle of the turbine and is a key determinant of efficiency and power. The efficiency of a gas turbine improves as the maximum cycle temperature increases, as expressed in the formula provided in the solution steps. The relationship is such that the higher the maximum cycle temperature, the greater the potential thermal efficiency.

To put it in culinary terms, it's akin to the flame setting under a boiling pot of water – the higher the flame, the quicker the water reaches boiling point, converting more water to steam for the same amount of heat (or energy) input. But just as boiling water can cause wear and tear on your pot, higher temperatures can strain a turbine's materials. The exercise underlines the theoretical impact of this temperature, but material limitations impose practical upper bounds on the maximum temperature that can be used in practice.