Question

If the inlet state and the exit pressure are specified for a twostage turbine with reheat between the stages and operating at steady state, show that the maximum total work output is obtained when the pressure ratio is the same across each stage. Use a cold air-standard analysis assuming that each compression process is isentropic, there is no pressure drop through the reheater, and the temperature at the inlet to each turbine stage is the same. Kinetic and potential energy effects can be ignored.

Short answer

The maximum total work output is achieved when the pressure ratio across each stage is equal, i.e., \( r_1 = r_2 = \sqrt{\frac{P_1}{P_3}} \.

Step-by-step solution

- State the Given Information

The problem provides the following information: inlet state and exit pressure of a two-stage turbine, reheat between stages, steady state operation, isentropic compression, no pressure drop through the reheater, and equal temperature at the inlet to each turbine stage. Kinetic and potential energy effects can be ignored.

- Define the Pressure Ratio

Let the inlet pressure be denoted as \( P_1 \) and the final exit pressure as \( P_3 \). The intermediate pressure after the first turbine stage and reheater is \( P_2 \). The pressure ratios across each turbine stage are \( r_1 = \frac{P_1}{P_2} \) and \( r_2 = \frac{P_2}{P_3} \).

- Isentropic Process Relationships

For an isentropic process, the temperature and pressure relationship can be described using the equation \ T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{\gamma - 1}{\gamma}}, \ where \( \gamma \) is the specific heat ratio. Similarly for the second stage, \ T_3 = T_1 \left(\frac{P_3}{P_2}\right)^{\frac{\gamma - 1}{\gamma}}.

- Total Work Output

The total work output involves the work done by each turbine stage. Using the fact that the temperature at the inlet to each turbine stage is the same and that each stage is isentropic, the work output will be maximized if each stage has the same pressure ratio. Thus, \( P_2 \) will be the geometric mean of \( P_1 \) and \( P_3 \), i.e., \( P_2 = \sqrt{P_1 \cdot P_3} \). Hence, \( r_1 = r_2 = \sqrt{\frac{P_1}{P_3}} \).

- Conclusion

When the pressure ratio is the same across each stage, the maximum total work output is achieved, fulfilling the given requirements.

Key concepts

isentropic process

An isentropic process is an idealized thermodynamic process that is both adiabatic (no heat transfer) and reversible. In simpler terms, it means that the system does not lose or gain heat, and the process can be reversed without any entropy change.
For an isentropic process involving a perfect gas, the relationship between temperature and pressure is described by the equation: \( T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{\gamma - 1}{\gamma}} \), where \( T \) is the temperature, \( P \) is the pressure, and \( \gamma \) is the specific heat ratio (ratio of specific heats at constant pressure and constant volume).
This relationship is crucial for understanding the behavior of gases in turbines and compressors. It helps determine the changes in temperature and pressure as the gas undergoes compression or expansion in the turbine stages.
To fully grasp this concept, remember:

• Isentropic processes are idealized and do not account for real-world inefficiencies.
• The specific heat ratio \( \gamma \) is typically around 1.4 for air.

pressure ratio

The pressure ratio in a thermodynamic process is the ratio of the pressure at the beginning and end of the process. In the context of a two-stage turbine, there are two pressure ratios to consider:

• First stage: \( r_1 = \frac{P_1}{P_2} \), where \( P_1 \) is the initial pressure and \( P_2 \) is the intermediate pressure after the first stage.
• Second stage: \( r_2 = \frac{P_2}{P_3} \), where \( P_2 \) is the intermediate pressure and \( P_3 \) is the final exit pressure.

The key to maximizing the work output of a two-stage turbine is to make these pressure ratios equal, meaning \( r_1 = r_2 \). This condition ensures that each stage of the turbine operates at its most efficient pressure ratio, resulting in maximum overall efficiency.
In mathematical terms, the ideal intermediate pressure \( P_2 \) is the geometric mean of \( P_1 \) and \( P_3 \): \[ P_2 = \sqrt{P_1 \cdot P_3} \]
This geometric mean balances the pressures across both stages, optimizing the turbine's performance.

reheat between stages

Reheating is a technique used in multi-stage turbines to heat the working fluid between stages. This process improves the efficiency of the turbine by ensuring that the fluid entering each stage is at a higher temperature.
In a two-stage turbine, reheating occurs between the first and second stages. The reheat increases the temperature of the working fluid, which often leads to higher work output from the turbine. Here are some key points about reheat:

• Reheating helps maintain a higher temperature, which improves thermal efficiency.
• It helps to prevent excessive cooling of the working fluid, which can reduce the efficiency of subsequent stages.
• In the context of our problem, it is assumed there is no pressure drop through the reheater, meaning the pressure remains constant during reheating.

This technique is commonly used in steam turbines in power plants, as it helps to maximize power output and efficiency.

steady state operation

Steady state operation occurs when the variables defining the state of a system (temperature, pressure, etc.) do not change over time. In a steady-state process, any given property of the working fluid remains approximately constant.
This is an important assumption for analyzing complex systems like turbines because it simplifies the calculations. It allows us to focus on the relationships between different stages without worrying about transient effects.
In our two-stage turbine problem:

• The steady state assumption ensures that the mass flow rate and energy flow rate are constant through the stages.
• This assumption simplifies the computation of work output and makes it easier to apply the isentropic process relationships.
• It allows us to ignore kinetic and potential energy effects, focusing solely on pressures and temperatures.

Remember, steady state means predictable and unchanging operation conditions, which is ideal for theoretical analysis and helps in achieving simplified and accurate solutions.

cold air-standard analysis

Cold air-standard analysis is a simplification used in thermodynamic analysis of air-based cycles, like those in gas turbines. This analytical approach assumes:

• The working fluid (air) behaves as an ideal gas.
• The specific heats \( c_p \) and \( c_v \) are constant throughout the process.
• The reference temperature is considered to be approximately constant, often at a lower value (hence 'cold' air).

These assumptions make it easier to perform calculations and predict performance. However, they may introduce some inaccuracies because in real life, specific heats vary with temperature.
The cold air-standard analysis is particularly helpful when studying cycles like the Brayton cycle, which is often used to describe the operation of gas turbines. By assuming air behaves ideally and properties like \( \gamma \) (specific heat ratio) remain constant, the analysis becomes significantly more straightforward.
Despite its simplicity, this form of analysis provides valuable insights and is frequent in educational settings to teach fundamental concepts before moving on to more complex, realistic models.