Question

Water is the working fluid in an ideal Rankine cycle. The condenser pressure is \(8 \mathrm{kPa}\), and saturated vapor enters the turbine at (a) \(18 \mathrm{MPa}\) and (b) \(4 \mathrm{MPa}\). The net power output of the cycle is \(100 \mathrm{MW}\). Determine for each case the mass flow rate of steam, in \(\mathrm{kg} / \mathrm{h}\), the heat transfer rates for the working fluid passing through the boiler and condenser, each in \(\mathrm{kW}\), and the thermal efficiency.

Short answer

Use steam tables to find enthalpy values for different states. Calculate the net work output, heat addition, and rejection for each case. Solve for mass flow rate, heat transfer rates, and thermal efficiency.

Step-by-step solution

Identify Given Data

Condenser pressure: 8 kPaCase (a): Turbine inlet pressure: 18 MPaCase (b): Turbine inlet pressure: 4 MPaNet power output: 100 MW

Determine State Points (a)

Using steam tables or Mollier chart, find the properties at each state:1. Saturated vapor at 18 MPa (inlet to turbine)2. Exit of turbine (8 kPa, determine quality if needed using isentropic process)3. Saturated liquid at 8 kPa (inlet to pump)4. Exit of pump (8 kPa to 18 MPa)Calculate the specific enthalpies and entropies accordingly.

Determine State Points (b)

Using steam tables or Mollier chart, find the properties at each state:1. Saturated vapor at 4 MPa (inlet to turbine)2. Exit of turbine (8 kPa)3. Saturated liquid at 8 kPa (inlet to pump)4. Exit of pump (8 kPa to 4 MPa)Calculate the specific enthalpies and entropies for this case.

Calculate Work and Heat Transfers (a)

Calculate the work done by the turbine and the pump input work. Use the specific enthalpies:o Turbine work:\[ W_t = h_1 - h_2 \]o Pump work (assuming incompressible liquid):\[ W_p = v_f(P_2 - P_1) \]o Net work output:\[ W_{net} = W_t - W_p \]Thermal efficiency:\[ \eta = \frac{W_{net}}{Q_{boiler}} \]Heat added in the boiler:\[ Q_{boiler} = h_1 - h_4 \]Heat rejected in the condenser:\[ Q_{condenser} = h_2 - h_3 \]

Calculate Work and Heat Transfers (b)

Repeat the calculations in Step 4 for Case (b) using the corresponding enthalpies for 4 MPa inlet pressure.

Determine Mass Flow Rate for Case (a)

Using the net power output \[ \dot{m} = \frac{Power}{W_{net}} \]Convert to \mathrm{kg/h} by multiplying by 3600.

Determine Mass Flow Rate for Case (b)

Using the net power output \[ \dot{m} = \frac{Power}{W_{net}} \]Convert to \mathrm{kg/h} by multiplying by 3600.

Finalize all Results

Compile all values including mass flow rates, heat transfer rates for both cases, and thermal efficiencies.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature as well as their relation to energy and work. In thermodynamics, we study systems and how they interact with their surroundings. An important principle in thermodynamics is the conservation of energy, which states that energy cannot be created or destroyed, only transformed or transferred. This is the foundation for analyzing cycles such as the Rankine cycle.
In an ideal Rankine cycle, water undergoes a series of processes, including heating, expansion, cooling, and compression. Understanding these processes and applying thermodynamic principles helps determine key cycle parameters, such as work output and thermal efficiency.

Steam Tables

Steam tables are essential tools in thermodynamics, providing vital data for water and steam properties at different temperatures and pressures. They typically include information like enthalpy (h), entropy (s), specific volume (v), and internal energy (u). These tables are used to find exact state properties for processes involving steam and water. For instance:
- Saturated vapor properties at a specific pressure
- Superheated steam properties at a specific temperature and pressure
In our Rankine cycle problem, steam tables help us determine the enthalpies and entropies at various state points, which are necessary for calculating the cycle's performance metrics. The values from the steam tables allow us to determine the work done by the turbine and pump as well as the heat added in the boiler and removed in the condenser.

Isentropic Process

An isentropic process is a thermodynamic process that occurs at constant entropy. In other words, it is an idealized process that is both reversible and adiabatic (no heat transfer). Such processes are important in the analysis of thermodynamic cycles.
For the ideal Rankine cycle, the expansion in the turbine and the compression by the pump are often assumed to be isentropic. This simplifies calculations, as it allows us to use isentropic relations to find the final states of the steam.
For example, in the turbine, steam undergoes an isentropic process from high-pressure saturated vapor to a low-pressure state. The entropy remains constant throughout this process:

• Using the initial state entropy and steam tables, we can find the final state by looking at the saturation properties at the condensing pressure and determining the quality of steam at that point.

Thermal Efficiency

Thermal efficiency is a measure of how well a thermodynamic cycle converts heat into work. It is defined as the ratio of net work output to the heat input into the cycle. For the ideal Rankine cycle, we compute the thermal efficiency using the specific enthalpies found from the steam tables.
The formula for the thermal efficiency (\eta) of the Rankine cycle is:\[\begin{equation}\eta = \frac{W_{net}}{Q_{boiler}}\end{equation}\]where:

• \(W_{net}\) is the net work output (turbine work minus pump work).
• \(Q_{boiler}\) is the heat added in the boiler.

Knowing the net power output of the cycle, we can then determine the mass flow rate of steam required. This will help us in understanding the overall effectiveness and performance of the power plant.Evaluating thermal efficiency provides insights into how improvements can be made to enhance cycle performance, such as by increasing boiler pressure or superheating steam to higher temperatures.