Question

A power plant operates on a regenerative vapor power cycle with one open feedwater heater. Steam enters the first turbine stage at \(12 \mathrm{MPa}, 520^{\circ} \mathrm{C}\) and expands to \(1 \mathrm{MPa}\), where some of the steam is extracted and diverted to the open feedwater heater operating at \(1 \mathrm{MPa}\). The remaining steam expands through the second turbine stage to the condenser pressure of \(6 \mathrm{kPa}\). Saturated liquid exits the open feedwater heater at \(1 \mathrm{MPa}\). For isentropic processes in the turbines and pumps, determine for the cycle (a) the thermal efficiency and (b) the mass flow rate into the first turbine stage, in \(\mathrm{kg} / \mathrm{h}\), for a net power output of \(330 \mathrm{MW}\).

Short answer

Thermal efficiency can be calculated using the enthalpy differences from each state. The mass flow rate into the first turbine is derived based on a net power output of 330 MW using the energy balances.

Step-by-step solution

Identify state points and their properties

Determine the properties of steam at each state point using steam tables or Mollier charts. State points are: (1) Steam entering the high-pressure turbine (12 MPa, 520°C), (2) Steam exiting the high-pressure turbine and entering the open feedwater heater (1 MPa), (3) Steam exiting the open feedwater heater (saturated liquid at 1 MPa), (4) Steam entering the low-pressure turbine (isentropic process from state 2 to 0.006 MPa), and (5) Condensate from the condenser (saturated liquid at 0.006 MPa).

Calculate enthalpies at each state point

Using the properties identified in Step 1, find the enthalpies (h1, h2, h3, h4, h5): h1 = Enthalpy at (12 MPa, 520°C) h2 = Enthalpy at (1 MPa post isentropic expansion from 12 MPa) h3 = Enthalpy of saturated liquid at 1 MPa h4 = Enthalpy at 6 kPa post isentropic expansion from 1 MPa h5 = Enthalpy of saturated liquid at 6 kPa.

Determine mass flow rates using energy balances

Using conservation of mass and energy principles, find the mass flow rates through each stage. Let the mass flow rate entering the first turbine be m1. The mass flow rate at the feedwater heater (extracted steam) is part of m1. Use the power output and energy balances to find the relationships between these mass flow rates.

Calculate the thermal efficiency

Thermal efficiency, \(\frac{W_{net}}{Q_{in}}\), can be defined as \(\frac{(h1-h2) + (1-y)(h2-h4)}{h1-h3}\), where \(y\) is the fraction of steam extracted. Identify the values of work from each turbine and the heat added to the boiler, and use these values to form the efficiency equation.

Find the mass flow rate for 330 MW net power output

Using the net power output and the enthalpy differences per unit mass flow (from Steps 1 and 2), find the mass flow rate using the formula: \(m = \frac{330 \times 10^6}{W_{turbine} - W_{pump}}\). Adjust this for unit consistency (e.g., converting to kg/h).

Key concepts

Thermal Efficiency

Thermal efficiency is a measure of how well an energy conversion process transforms input energy into useful output energy. In a regenerative vapor power cycle, it denotes the ratio of net work output to the heat input. For thermal efficiency calculation in this context, use the formula:

\(\frac{W_{net}}{Q_{in}})\).
Net work output (\(W_{net}\)) is the total work produced by the turbines minus the work consumed by the pumps. Heat input (\(Q_{in}\)) is the energy added to the steam in the boiler.

For this cycle, we compute the work done by each turbine and subtract the work done by the pumps. Then, divide it by the heat added in the boiler to compute the efficiency. Breaking down the terms can simplify the calculation:

• Numerator: Work done per unit mass in each turbine stage and pumps.
• Denominator: Enthalpy increase from feedwater to high-pressure steam in the boiler.

Mass Flow Rate

The mass flow rate represents the amount of mass passing through a system per unit of time. In vapor power cycles, it is crucial for determining energy exchange rates. To find it, we use energy balances for each component (like turbines and pumps).

For this problem, consider the extraction and expansion processes:

• Identify enthalpy values (energy per unit mass) and differences at various state points.
• Calculate the mass flow rate into the first turbine using: \(m = \frac{330 \times 10^6}{W_{turbine} - W_{pump}}\).
• Adjust this value to kg/h for practical purposes.

Accurate flow rates ensure the system's energy needs and outputs align correctly.

Isentropic Process

An isentropic process is a thermodynamic process that occurs without any entropy change or heat transfer. It ideally represents reversible, adiabatic processes often assumed for turbines and pumps in power cycles.

For analysis:

• Determine state points and their properties assuming isentropic expansion or compression.
• Use steam tables to find the specific enthalpy and entropy values at these points.
• Throughout the turbines, apply the principle: \(s_2 = s_3\) for no entropy change from the high-pressure to low-pressure stages.

Helps simplify calculations at various stages (before and after expansion).

Steam Tables

Steam tables present properties of water and steam, listing relationships like enthalpy, entropy, temperature, and pressure. They are indispensable for analyzing vapor power cycles.

When using steam tables:

• Identify state points and retrieve corresponding properties such as \(h, s, \text{ and }v\).
• Calculate work and heat interactions in cycles by referring to these tabulated values.
• For this problem: Use properties at 12 MPa and 520°C, 1 MPa, and at condenser pressure (0.006 MPa).

Steam tables ease complex calculations by providing ready-to-use thermodynamic data.

Energy Balance

Energy balance involves accounting for energy entering and leaving each component in the cycle. For a regenerative vapor power setup:

Apply the principle of conservation of energy by matching input and output energies:

• In turbines: Balance the initial and final enthalpies, adjusting for extracted steam's role.
• In feedwater heaters: Ensure energy added by extracted steam renders the feedwater reaching saturation conditions.
• For pumps: Consider the energy added to feedwater in pumping processes.

Accurate energy balances ensure precise performance calculations for efficiency and output metrics.