Question

Consider a regenerative vapor power cycle with two feedwater heaters, a closed one and an open one. Steam enters the first turbine stage at \(8 \mathrm{MPa}, 480^{\circ} \mathrm{C}\), and expands to \(2 \mathrm{MPa}\). Some steam is extracted at \(2 \mathrm{MPa}\) and fed to the closed feedwater heater. The remainder expands through the second-stage turbine to \(0.3 \mathrm{MPa}\), where an additional amount is extracted and fed into the open feedwater heater, which operates at \(0.3 \mathrm{MPa}\). The steam expanding through the third-stage turbine exits at the condenser pressure of \(8 \mathrm{kPa}\). Feedwater leaves the closed heater at \(205^{\circ} \mathrm{C}, 8 \mathrm{MPa}\), and condensate exiting as saturated liquid at \(2 \mathrm{MPa}\) is trapped into the open heater. Saturated liquid at \(0.3 \mathrm{MPa}\) leaves the open feedwater heater. The net power output of the cycle is \(100 \mathrm{MW}\). If the turbine stages and pumps are isentropic, determine (a) the thermal efficiency. (b) the mass flow rate of steam entering the first turbine, in \(\mathrm{kg} / \mathrm{h}\).

Short answer

Thermal efficiency calculated from work and heat transfers. Mass flow rate from net power divided by steam work.

Step-by-step solution

Analyzing the Turbine Expansion Stages

First stage: Steam enters at 8 MPa, 480°C and expands to 2 MPa. Extract steam at 2 MPa for closed feedwater heater. Second stage: Remaining steam expands from 2 MPa to 0.3 MPa, where further steam is extracted for open feedwater heater. Third stage: Steam finally expands to condenser pressure of 8 kPa.

Determining State Properties at Key Points

Use steam tables or Mollier diagram to determine the properties (enthalpy, entropy) at different states (1 to 10) based on given pressures and temperatures. Specifically, for points 1 (8 MPa, 480°C), 2 (2 MPa), 3 (0.3 MPa), 4 (8 kPa).

Calculation of Work and Heat Transfer

Calculate work done by each turbine stage using enthalpy changes (1 to 2, 2 to 3, 3 to 4). Calculate work done by pumps using enthalpy changes for feedwater heaters.

Efficiency Calculation

Calculate the total heat added to the system. The thermal efficiency is given by the net work output divided by the heat added.

Determining Mass Flow Rate

Using the given net power output (100 MW) and calculated turbine work per kg of steam, determine the mass flow rate by dividing the net power output by the specific work done by the steam (consider all turbine stages).

Key concepts

Thermal Efficiency

Thermal efficiency is a key measure of performance in a regenerative vapor power cycle. It's defined as the ratio of net work output to the total heat input. In simpler terms, it tells you how much of the heat energy added to the system is converted to useful work.
In this cycle, the process involves stages where steam enters the turbine, expands, and gets reheated. The efficiency in each stage contributes to the overall efficiency. Here’s a quick formula to help:
\[ \text{Thermal Efficiency} (\text{η}) = \frac{W_{net}}{Q_{in}} \times 100 \text{(%)}, \ \text{where,} \ W_{net} = \text{Net work done by the cycle} \ Q_{in} = \text{Total heat added to the system} \]\ In regenerative cycles, using feedwater heaters improves the thermal efficiency by preheating the feedwater before it returns to the boiler. This means less fuel is used to heat the water, improving overall efficiency.

Feedwater Heater

Feedwater heaters are devices used to preheat the water entering the boiler using steam extracted from intermediate stages of a turbine. They come in two types: closed feedwater heaters and open feedwater heaters.

• Closed Feedwater Heater: This heater allows steam to transfer its heat to the feedwater without mixing with it.


• Open Feedwater Heater: In this heater, extracted steam mixes directly with the feedwater.

Benefit of using feedwater heaters is that they increase the cycle's thermal efficiency. Preheating the feedwater requires less fuel to heat the water to steam, which means the boiler works less hard.
In the given problem, steam at 2 MPa is extracted and fed into a closed feedwater heater. Another portion is extracted at 0.3 MPa and fed into an open feedwater heater. The final heating occurs before the water reaches the boiler.

Turbine Expansion

Turbine expansion in a regenerative vapor power cycle occurs in multiple stages. Each stage involves steam expanding through the turbine and performing work.

• First Stage: Steam expands from 8 MPa, 480°C to 2 MPa, with some of the steam extracted for the closed feedwater heater.


• Second Stage: The remaining steam further expands from 2 MPa to 0.3 MPa. Additional steam is extracted for the open feedwater heater.


• Third Stage: The remaining steam finally expands to 8 kPa, exiting the turbine at condenser pressure.

At each stage, the extracted steam helps preheat the feedwater, contributing to greater overall efficiency. The energy conversion during these expansions is quantified using enthalpy changes, found in steam tables or Mollier diagrams, and is crucial for calculating overall work done.