Question

In the preliminary design of a power plant, water is chosen as the working fluid and it is determined that the turbine inlet temperature may not exceed \(520^{\circ} \mathrm{C}\). Based on expected cooling water temperatures, the condenser is to operate at a pressure of \(0.06\) bar. Determine the steam generator pressure required if the isentropic turbine efficiency is \(80 \%\) and the quality of steam at the turbine exit must be at least \(90 \%\).

Short answer

Determine the steam generator pressure by iterating the inlet pressure until the exit quality constraint is satisfied for a turbine exit with a quality of at least 90% and an isentropic efficiency of 80%.

Step-by-step solution

Identify State Points and Property Values

At the turbine inlet, the temperature is given as \(520^{\circ}C\) and the pressure needs to be determined. At the turbine exit, the pressure is given as \(0.06\) bar and the quality must be at least \(90\%\) (\(x_2 = 0.9\)). Let's denote the turbine inlet state as point 1 (\(h_1\)) and the exit state as point 2 (\(h_2\)).

Use Steam Tables for State Points

Using steam tables to find the properties at the turbine inlet (state 1) and the condenser (state 2). For exit pressure, \(P_2 = 0.06\) bar, find \(h_{f2}\) and \(h_{fg2}\), and the specific volumes. At inlet temperature \(T_1 = 520^{\circ}C\), determine \(h_1\) based on chosen pressure.

Isoentropic Efficiency Relation

The isentropic efficiency \(\eta_{is}\) is given as \(80\%\). The actual turbine work and enthalpy drop can be related as:\[ \eta_{is} = \frac{h_1 - h_2}{h_1 - h_{2,s}} \]Where \(h_{2,s}\) is the enthalpy at state 2 assuming an isentropic process.

Determine \(h_{2,s}\)

Assume an isentropic process (entropy remains constant), find \(s_2 = s_1\). Use steam tables to determine \(h_{2,s}\) at \(P_2 = 0.06\) bar.

Compute the Actual Exit Enthalpy

Using the isentropic efficiency and \(h_{2,s}\):\[ h_2 = h_1 - \eta_{is} (h_1 - h_{2,s}) \], verify that the quality \(x_2\) at this \(h_2\) is at least \(90\%\).

Adjust Inlet Pressure

Iterate the inlet pressure until the exit quality constraint (at least \(90\%\)) is satisfied. Adjust \(h_1\) with new pressure and re-calculate.

Final Calculations and Verification

Verify all calculations and ensure that the inlet pressure provides the necessary conditions for the outlet quality and the isentropic efficiency requirements.

Key concepts

Isentropic Efficiency

Isentropic efficiency is a measure of the effectiveness of a thermodynamic device, such as a turbine or compressor, in converting energy. It compares the actual performance of the device with an ideal process, where no energy loss due to friction or other inefficiencies occurs.
The formula for isentropic efficiency (\text{\textbackslash eta_{\textnormal{is}}}) is:\( \text{\textbackslash eta_{\textnormal{is}} = \frac{h_1 - h_2}{h_1 - h_{2,s}}} \)
Here:

• \text{\textbackslash h_1} is the enthalpy at the inlet,
• \text{\textbackslash h_2} is the actual enthalpy at the exit,
• and \text{\textbackslash h_{2,s}} is the enthalpy at the exit assuming it is an isentropic process.

Because real processes are not perfectly efficient, the actual enthalpy change (\text{\textbackslash h_1 - h_2}) will always be less than the ideal enthalpy change (\text{\textbackslash h_1 - h_{2,s}}), leading to efficiency values less than 100%. This efficiency impacts the power plant’s overall performance, hence optimizing it is essential for better energy conversion and cost-effectiveness.

Steam Quality

Steam quality refers to the proportion of vapor in a mixture of vapor and liquid at a specific pressure and temperature. It is usually expressed as a percentage. High-quality steam, close to 100%, is primarily vapor, which is ideal for turbine performance and reduces erosion and mechanical wear.
The quality of the steam (\text{\textbackslash x}) can be found using the following relationship:\( \text{\textbackslash x = \frac{m_{\textnormal{vapor}}}{m_{\textnormal{total}}}} \)
For the turbine exit condition in this exercise, the quality (\text{\textbackslash x_2}) needs to be at least 90% to ensure that most of the steam is vapor:

• \text{\textbackslash x \rightarrow \frac{M_{\textnormal{vapor}}}{M_{\textnormal{total}}}} = 0.9

Ensuring this high steam quality minimizes the risk of turbine blade damage and enhances overall efficiency. Low-quality steam can cause problems such as corrosion and inefficiencies in the cycle.

Thermodynamic Cycles

Thermodynamic cycles are processes that involve a series of thermodynamic steps returning a system to its initial state, creating a loop. These cycles are fundamental in the operation of power plants, refrigerators, and engines.
The most common cycle used in power plant design is the Rankine cycle, which involves four main processes:

• 1. **Isentropic Compression** in a pump,
• 2. **Constant Pressure Heat Addition** in a boiler,
• 3. **Isentropic Expansion** in a turbine,
• 4. **Constant Pressure Heat Rejection** in a condenser.

In this exercise, water flows through these stages, undergoing various thermodynamic transformations. The working fluid is compressed in a pump, heated in a boiler, expanded in a turbine, and finally condensed in a condenser, thus completing the cycle. Adjusting the parameters (like inlet pressure or condenser pressure) can optimize the cycle for better efficiency, as seen in the problem's iterative process for determining the ideal pressure to maintain high steam quality and efficient energy conversion.