Question

A power plant operates on a regenerative vapor power cycle with one closed feedwater heater. Steam enters the first turbine stage at 120 bar, \(520^{\circ} \mathrm{C}\) and expands to 10 bar, where some of the steam is extracted and diverted to a closed feedwater heater. Condensate exiting the feedwater heater as saturated liquid at 10 bar passes through a trap into the condenser. The feedwater exits the heater at 120 bar with a temperature of \(170^{\circ} \mathrm{C}\). The condenser pressure is \(0.06\) bar. For isentropic processes in each turbine stage and the pump, determine for the cycle (a) the thermal efficiency and (b) the mass flow rate into the first-stage turbine, in \(\mathrm{kg} / \mathrm{h}\), if the net power developed is \(320 \mathrm{MW}\).

Short answer

Thermal efficiency and mass flow rate are evaluated using specific enthalpies, energy balances, and the net power output.

Step-by-step solution

- Determine the specific enthalpies

Using steam tables, determine the specific enthalpies at all key points in the cycle. Identify the enthalpies at turbine entrance, after the first stage of expansion, after condensation, and after heating in the closed feedwater heater.

- Calculate isentropic enthalpies

Using the isentropic assumption, find the enthalpies for the process in the turbines and pumps. For instance, the enthalpy after isentropic expansion from 120 bar and 520°C to 10 bar can be found.

- Energy balance equations

Set up energy balance equations for the different components of the cycle: turbines, feedwater heater, condenser, and pump. Express the net power output, heat added, and heat rejected in terms of the specific enthalpies and mass flow rates.

- Thermal efficiency

Calculate the thermal efficiency using the formula \(\text{efficiency} = \frac{\text{Net Work Output}}{\text{Heat Input}}\). Evaluate the net work output and heat input based on the enthalpy values determined in Step 1.

- Mass flow rate calculation

Using the net power output of 320 MW and the relations from previous steps, calculate the mass flow rate into the turbine. This involves using the formula \(\text{Power} = \text{mass flow rate} \times \text{work per unit mass}\).

Key concepts

Thermal Efficiency

Thermal efficiency measures how well a power cycle converts heat energy into mechanical energy. In a regenerative vapor power cycle, thermal efficiency is key to understanding overall performance.
First, we determine specific enthalpies using steam tables at various points in the cycle. This includes the turbine entrance, post-expansion, and after condenser and feedwater heating.
For our cycle, thermal efficiency can be found using: \[ \text{efficiency} = \frac{\text{Net Work Output}}{\text{Heat Input}} \]
Here, Net Work Output is the difference between turbine work output and the pump work. Heat Input is the enthalpy change during heating in the boiler.

Mass Flow Rate

Mass flow rate is crucial for understanding how much steam is processed in a power cycle. To calculate it, we use the net power output and specific work per mass. For our case with a net power output of 320 MW, we use: \[ \text{Power} = \text{mass flow rate} \times \text{work per unit mass} \]
By rearranging, we get: \[ \text{mass flow rate} = \frac{\text{Power}}{\text{work per unit mass}} \]
This formula tells us the mass flow rate in \text{kg/}text{h} needed to generate the desired power output.

Entropy Calculations

Entropy calculations help us track the energy's disorder and ensure the process stays within the second law of thermodynamics.

In isentropic (constant entropy) processes, changes in system parameters like pressure and temperature can be detailed using steam tables. Calculate entropy at the turbine entrance and throughout the expansion process. This aids in finding isentropic enthalpies and understanding energy distribution.

Ensure no entropy is generated for truly isentropic processes. If entropy increases, it indicates inefficiencies or losses.

Isentropic Process

An isentropic process is an ideal process with constant entropy. It's a critical assumption for simplifying calculations in turbines and pumps.
For turbines, we can find the specific enthalpies during expansion using isentropic relations. This means that if steam enters a turbine at high pressure and temperature, we can predict its exit state based solely on pressure, assuming no entropy change.
For pumps, similar calculations apply. Isentropic efficiency is used in real-world scenarios where processes deviate from the ideal case.

Energy Balance Equations

Energy balance equations are fundamental to analyzing power cycles. Each component – turbines, feedwater heater, condenser, and pump – must have its energy intake and output accounted for.
For turbines, the work output is derived from the change in enthalpy: \[ W = m (\text{h}_{\text{in}} - \text{h}_{\text{out}}) \]
Feedwater heaters use energy balances to determine the heat added to the fluid…
Condensers capture the heat rejected to the environment…
The pump energy balance shows the work required to pressurize the fluid, affecting net power.
Analyzing these balances helps understand efficiency and system losses.