Question

Superheated steam at \(18 \mathrm{MPa}, 560^{\circ} \mathrm{C}\), enters the turbine of a vapor power plant. The pressure at the exit of the turbine is \(0.06\) bar, and liquid leaves the condenser at \(0.045\) bar, \(26^{\circ} \mathrm{C}\). The pressure is increased to \(18.2 \mathrm{MPa}\) across the pump. The turbine and pump have isentropic efficiencies of 82 and \(77 \%\), respectively. For the cycle, determine (a) the net work per unit mass of steam flow, in \(\mathrm{kJ} / \mathrm{kg}\). (b) the heat transfer to steam passing through the boiler, in kJ per \(\mathrm{kg}\) of steam flowing. (c) the thermal efficiency. (d) the heat transfer to cooling water passing through the condenser, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of steam condensed.

Short answer

The steps involve determining enthalpies at various points, adjusting for efficiencies, and calculating net work, heat transfer, and efficiency.

Step-by-step solution

- Analyze Turbine Work per Unit Mass

Determine the state properties at the entrance and exit of the turbine. Use steam tables to find the enthalpies and entropies of the superheated steam at 18 MPa and 560°C. The properties at the exit (0.06 bar) can be determined assuming isentropic expansion.

- Calculate Turbine Work with Isentropic Efficiency

Use the efficiency definition to adjust the turbine work output. First, calculate the work assuming isentropic conditions and then apply the efficiency: \[ W_{turbine} = \text{Eff_{turbine}} \times (h_1 - h_{2s}) \]

- Determine Pump Work per Unit Mass

Analyze the pump state's properties at the entrance (0.045 bar, 26°C) and determine pump work for increasing the pressure to 18.2 MPa. Apply isentropic efficiency to get actual pump work: \[ W_{pump} = \frac{W_{pump,isen}}{\text{Eff_{pump}}} \]

- Calculate Net Work per Unit Mass

Subtract the pump work from the turbine work: \[ W_{net} = W_{turbine} - W_{pump} \]

- Evaluate Heat Transfer in the Boiler per Unit Mass

Find the enthalpy before and after the boiler by evaluating the steam conditions using steam tables. The heat added in the boiler corresponds to the enthalpy difference: \[ q_{in} = h_1 - h_{cond} \]

- Determine Thermal Efficiency

The thermal efficiency is calculated by dividing the net work by the heat added: \[ \text{Efficiency} = \frac{W_{net}}{q_{in}} \]

- Evaluate Heat Transfer in the Condenser per Unit Mass

The heat rejected in the condenser can be calculated by finding the enthalpy difference as the steam exits the condenser (0.045 bar, liquid). \[ q_{out} = h_{2} - h_{cond} \]

Key concepts

Isentropic Efficiency

In a vapor power plant, understanding isentropic efficiency is crucial for analyzing the performance of turbines and pumps. Isentropic efficiency measures how closely a real process (like in a turbine or a pump) approaches an ideal isentropic process (a process with no entropy change). For the turbine, isentropic efficiency \(\text{Eff}_{\text{turbine}}\) can be calculated as follows: \[ \text{Eff}_{\text{turbine}} = \frac{h_1 - h_{2\text{s}}}{h_1 - h_2} \] Here, \(h_1\) is the enthalpy at the turbine inlet, \(h_{2\text{s}}\) is the enthalpy at the turbine outlet in the isentropic case, and \(h_2\) is the actual enthalpy at the turbine outlet. Similarly, for a pump, the isentropic efficiency \(\text{Eff}_{\text{pump}}\) is given by: \[ \text{Eff}_{\text{pump}} = \frac{W_{\text{pump,isen}}}{W_{\text{pump}}} \] Here, \(W_{\text{pump,isen}}\) is the work input under isentropic conditions, and \(W_{\text{pump}}\) is the actual work input. These efficiencies are always less than 100%, indicating losses due to factors like friction and heat transfer.

Turbine Work Calculation

Calculating the work done by a turbine starts with using steam tables to determine the enthalpy values at the inlet and outlet. For the given example, we need the enthalpy \(h_1\) at 18 MPa and 560°C. Assuming isentropic expansion, we find \(h_{2\text{s}}\) at 0.06 bar. The actual turbine work comes from adjusting for isentropic efficiency: \[ W_{\text{turbine}} = \text{Eff}_{\text{turbine}} \times (h_1 - h_{2\text{s}}) \] By substituting the efficiency of 82% \( (\text{Eff}_{\text{turbine}} = 0.82)\), we get: \[ W_{\text{turbine}} = 0.82 \times (h_1 - h_{2\text{s}}) \] This formula accounts for the losses and gives the usable turbine work.

Pump Work Calculation

Pumps in vapor power plants increase the pressure of the fluid, requiring work input. To find the work needed, start by determining the enthalpy at the initial state (0.045 bar, 26°C) using steam tables. Calculate the ideal isentropic pump work (\text{\(W_{\text{pump,isen}})\) for raising the pressure to 18.2 MPa. This work is adjusted with the isentropic efficiency to find the actual work required: \[ W_{\text{pump}} = \frac{W_{\text{pump,isen}}}{\text{Eff}_{\text{pump}}} \] Given an efficiency \( \text{Eff}_{\text{pump}} \) of 77% (\text{0.77}), we have: \[ W_{\text{pump}} = \frac{W_{\text{pump,isen}}}{0.77} \] This calculation helps in understanding the energy required to operate the pump, accounting for real-world inefficiencies.

Thermal Efficiency

Thermal efficiency measures how well the power plant converts heat into work. It's defined as the ratio of net work output to the heat input: \[ \text{Thermal Efficiency} = \frac{W_{\text{net}}}{q_{\text{in}}} \] We already calculated the net work per unit mass \(W_{\text{net}}\): \[ W_{\text{net}} = W_{\text{turbine}} - W_{\text{pump}} \] For heat added in the boiler \( (q_{\text{in}})\), we use the enthalpy difference before and after the boiler: \[ q_{\text{in}} = h_1 - h_{\text{cond}} \] Thermal efficiency aids in evaluating power plant performance, showing how effectively the plant converts fuel energy into useful work.

Heat Transfer in Boiler

The boiler in a vapor power plant adds energy to the steam, necessary for powering the turbine. To find the heat transfer per unit mass: \[ q_{\text{in}} = h_1 - h_{\text{cond}} \] Here, \( h_1 \) is the enthalpy at the boiler exit, and \( h_{\text{cond}} \) is the enthalpy of liquid leaving the condenser. The steam tables are essential because they provide these enthalpy values. By subtracting the initial enthalpy from the final enthalpy, we obtain the heat added in the boiler. This calculation evaluates the energy input required to produce steam.

Steam Tables

Steam tables are indispensable tools for analyzing vapor power plants. They provide thermodynamic properties of water and steam at various temperatures and pressures. Key properties include enthalpy, entropy, and specific volume. For instance, to solve the problem:

• Find enthalpy \( h_1 \) at 18 MPa and 560°C
• Determine enthalpy \( h_{\text{cond}} \) at 0.045 bar, 26°C
• Check enthalpy values for isentropic expansion \( h_{2s} \) at 0.06 bar

Using these tables helps us accurately perform calculations, ensuring we dimension the plant correctly and understand its performance. Hence, mastering how to read steam tables is fundamental for any engineering student.