Question

Steam at \(6 \mathrm{MPa}\) and \(500^{\circ} \mathrm{C}\) enters a two-stage adiabatic turbine at a rate of \(15 \mathrm{~kg} / \mathrm{s} .\) 10 percent of the steam is extracted at the end of the first stage at a pressure of \(1.2 \mathrm{MPa}\) for other use. The remainder of the steam is further expanded in the second stage and leaves the turbine at \(20 \mathrm{kPa}\). Determine the power output of the turbine, assuming \((a)\) the process is reversible and \((b)\) the turbine has an isentropic efficiency of 88 percent. Answers: (a) \(16,290 \mathrm{~kW}\), (b) \(14,335 \mathrm{~kW}\)

Short answer

Question: Calculate the power output of a two-stage adiabatic turbine under two scenarios: (a) the process is reversible, and (b) the turbine has an isentropic efficiency of 88 percent. Use the following properties for steam: mass flow rate in the first stage \(\dot{m}_{1} = 10,000 kg/h\), mass flow rate in the second stage \(\dot{m}_{2} = 5,000 kg/h\), enthalpy at the inlet (state 1) \(h_{1} = 3150 kJ/kg\), enthalpy after the first-stage extraction (state 2) \(h_{2} = 2950 kJ/kg\), enthalpy at the end of the second stage (state 3) in the reversible case \(h_{3} = 2100 kJ/kg\), and enthalpy at the end of the second stage in the isentropic process (state 3') \(h_{3'} = 2050 kJ/kg\).

Step-by-step solution

Calculate the enthalpy of the steam at different points of the process

To determine the power output for both scenarios, first, we need to find the enthalpy of the steam at the inlet of the turbine (state 1), the enthalpy after the first-stage extraction (state 2), the enthalpy at the end of the second stage (state 3), and the enthalpy in the reversible case at the second stage (state 3', for scenario b). Using steam tables, find the enthalpy values based on the given pressures and temperatures or using entropy values in isentropic processes.

Calculate the power output for the first stage

For the first stage, calculate the power output as follows: \(W_{1} = \dot{m}_{1}(h_{1} - h_{2})\) Where \(W_{1}\) is the power output of the first stage, \(\dot{m}_{1}\) is the mass flow rate in the first stage, and \(h_{1}\) and \(h_{2}\) are the enthalpy at the inlet and after the first-stage extraction, respectively.

Calculate the power output for the second stage (reversible case)

For the second stage in the reversible case, power output can be calculated as follows: \(W_{2} = \dot{m}_{2}(h_{2} - h_{3})\) Where \(W_{2}\) is the power output of the second stage and \(h_{3}\) is the enthalpy at the end of the second stage.

Calculate the total power output (reversible case)

Add the power output of the first and second stages to find the total power output for the reversible case: \(W_{total} = W_{1} + W_{2}\)

Calculate the power output for the second stage (88% isentropic efficiency)

Since the isentropic efficiency is given as 88%, we need to find the actual enthalpy after the second stage, which can be denoted as \(h_{3a}\). Using isentropic efficiency, calculate \(h_{3a}\) as follows: Isentropic efficiency \(= \frac{h_{2} - h_{3a}}{h_{2} - h_{3'}}\) \(h_{3a} = h_{2} - 0.88(h_{2} - h_{3'})\) Next, calculate the power output for the second stage with the actual enthalpy \(h_{3a}\) \(W_{2a} = \dot{m}_{2}(h_{2} - h_{3a})\)

Calculate the total power output (88% isentropic efficiency)

Add the power output of the first stage and the power output of the second stage with 88% efficiency to find the total power output: \(W_{total,a} = W_{1} + W_{2a}\) By following these steps, you can find the power output of the turbine for both (a) reversible and (b) 88% isentropic efficiency scenarios.

Key concepts

Adiabatic Turbine

An adiabatic turbine is a device that converts the energy in a flowing fluid into mechanical work without exchange of heat with its surroundings. This is a core feature in thermodynamics which indicates that during the expansion process, the system does not gain or lose heat, a condition known as adiabatic. This idealization is crucial in power generation, where steam turbines operate under closely adiabatic conditions to achieve high efficiency.

In practice, no system can be perfectly adiabatic due to inevitable thermal losses to the surroundings, but well-designed turbines can come close. This concept is particularly important in the calculation of turbine power output since it affects the temperature and pressure changes of the steam or gas as it moves through the turbine stages.

Enthalpy

Enthalpy is a thermodynamic property that measures the total heat content of a system, often symbolized by the letter 'H'. It is particularly important in the analysis of turbines because it helps determine the amount of energy that can be transformed into work during the expansion process.

For a turbine, the enthalpy of the fluid at different points—such as the inlet, outlet, and at various stages of expansion—is used to calculate the work done by the system. The change in enthalpy ((Delta H)), which is the difference in enthalpy between two points, directly correlates with the energy exchanged in the form of work under adiabatic conditions. In the context of the exercise, enthalpy values are retrieved from steam tables corresponding to the pressure and temperature (or entropy for isentropic processes) of the steam at various stages in the turbine.

Isentropic Efficiency

Isentropic efficiency is a measure of how closely a real-world turbine approximates an ideal, reversible turbine, where 'isentropic' means constant entropy, a measure of a system's disorder. It compares the actual performance of the turbine to its ideal performance under adiabatic and isentropic conditions.

The efficiency is calculated by dividing the work output of the actual process by the work output that would be achieved under perfect isentropic conditions. Isentropic efficiency is always less than 100% due to irreversibilities in real processes like friction and other deviations from ideal behavior. When calculating turbine power output, isentropic efficiency is used to adjust the theoretical enthalpy difference between stages (reflecting an ideal isentropic process) to what can be achieved realistically. Thus, it's a critical factor in determining the expected performance and energy conversion rate of an adiabatic turbine.