91Ó°ÊÓ

Water is the working fluid in an ideal regenerative Rankine cycle with one open feedwater heater. Superheated vapor enters the first-stage turbine at \(16 \mathrm{MPa}, 560^{\circ} \mathrm{C}\), and the condenser pressure is \(8 \mathrm{kPa}\). The mass flow rate of steam entering the first-stage turbine is \(120 \mathrm{~kg} / \mathrm{s}\). Steam expands through the first-stage turbine to \(1 \mathrm{MPa}\) where some of the steam is extracted and diverted to an open feedwater heater operating at \(1 \mathrm{MPa}\). The remainder expands through the second-stage turbine to the condenser pressure of \(8 \mathrm{kPa}\). Saturated liquid exits the feedwater heater at \(1 \mathrm{MPa}\). Determine (a) the net power developed, in \(\mathrm{kW}\). (b) the rate of heat transfer to the steam passing through the boiler, in \(\mathrm{kW}\). (c) the thermal efficiency. (d) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{s}\), if the cooling water undergoes a temperature increase of \(18^{\circ} \mathrm{C}\) with negligible pressure change in passing through the condenser.

Step by step solution

01

Determine Enthalpies

Use steam tables to find the enthalpies at each state in the cycle. Identify states: (1) exit of the boiler, (2) after the first-stage turbine, (3) exit of the open feedwater heater, (4) exit of the condenser, (5) after the feedwater pump, (6) after the second-stage turbine.

02

Calculate Work Done by Turbines

Use the enthalpies to calculate the work done by the first-stage and second-stage turbines. For the first stage: \[ W_{T1} = m \times (h_1 - h_2) \] For the second stage: \[ W_{T2} = m \times (h_3 - h_4) \] where: \(m\) is mass flow rate, and \(h_n\) are enthalpies at each state.

03

Calculate Pump Work

Calculate the work done by the pump for the feedwater. \[ W_P = \frac{V_x \times (P_y - P_z)}{\text{Efficiency}} \] where: \(V_x\) is the specific volume and \(P_y\) and \(P_z\) are pressures before and after the pump, respectively.

04

Net Work Developed

Combine the work from the turbines and subtract the work done by the pump. \[ W_{net} = (W_{T1} + W_{T2}) - W_P \]

05

Determine Heat Transfer in Boiler

Find the rate of heat transfer to the steam in the boiler using: \[ Q_{in} = m \times (h_1 - h_6) \]

06

Calculate Thermal Efficiency

Determine the thermal efficiency by combining the net work developed and the heat added: \[ \text{Thermal Efficiency} = \frac{W_{net}}{Q_{in}} \]

07

Mass Flow Rate of Cooling Water

Calculate the mass flow rate of the condenser cooling water using: \[ \text{Mass Flow Rate}_{CW} = \frac{Q_{out}}{c_p \times \text{Temperature Increase}} \] where \(Q_{out}\) is the heat rejected, and \(c_p\) is the specific heat capacity of water.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ideal Rankine cycle

The ideal Rankine cycle is a theoretical model for steam power plants. It provides a simplified framework to analyze how energy is converted into work through thermodynamic processes. The cycle consists of four main components:

• Boiler
• Turbine
• Condenser
• Pump

In this cycle, water is heated in the boiler to produce steam, which then expands in the turbine to generate work. The steam is subsequently condensed back into water in the condenser before being pumped back into the boiler to continue the cycle. By examining ideal Rankine cycles, we can understand how efficiency, enthalpy, and other thermodynamic properties are optimized in real power plants.

feedwater heater

A feedwater heater is used to preheat water before it enters the boiler. This process improves overall cycle efficiency. There are two types of feedwater heaters:

• Open Feedwater Heaters
• Closed Feedwater Heaters

In an open feedwater heater, partially expanded steam from the turbine mixes directly with feedwater. The mixture is at a higher temperature compared to water directly from the condenser. This reduces the energy required in the boiler to convert water into steam. In the given problem, after the first-stage turbine, steam expands to 1 MPa and is then extracted to the feedwater heater. The rest of the steam continues to expand to condenser pressure. Understanding feedwater heaters is crucial for grasping how thermal efficiency is enhanced in Rankine cycles.

thermal efficiency

Thermal efficiency measures how well a cycle converts heat into work. The formula for thermal efficiency for the Rankine cycle is: \ \text{Thermal Efficiency} = \frac{W_{net}}{Q_{in}} Here,

• \(W_{net}\) is the net work output.
• \(Q_{in}\) is the heat added to the system in the boiler.

To calculate this, you'll need the enthalpy values at different states and the work done by the pump and turbines. For the ideal Rankine cycle, higher efficiency is achieved by maximizing the work done by the turbine and minimizing heat and energy losses. By using feedwater heaters, less energy is required for heating, thus boosting efficiency.

enthalpy calculations

Enthalpy calculations are vital for analyzing the energy at various stages of the Rankine cycle. Enthalpy ( \(h\)) is a measure of the total energy content of the steam and water in the system. You can find these values in steam tables, which list the enthalpies at different temperatures and pressures. In the given problem, you need to determine enthalpies at key points:

• 
  The exit of the boiler
• After the first-stage turbine
• The exit of the open feedwater heater
• The exit of the condenser
• After the feedwater pump
• After the second-stage turbine

These enthalpy values help in calculating the work done by the turbines, the pump work, and the heat transfer in the boiler. Without accurate enthalpy calculations, it would be impossible to determine net work and thermal efficiency.

condenser cooling water

The condenser plays a crucial role in turning steam back into water before it re-enters the cycle. It relies on cooling water to remove the heat from the steam. In this problem, the cooling water undergoes a temperature increase of 18\textdegree\text{C}, meaning it absorbs a significant amount of heat. The mass flow rate of the cooling water can be calculated using: \ \text{Mass Flow Rate}_{CW} = \frac{Q_{out}}{c_p \times \text{Temperature Increase}} Where

• 
  \(Q_{out}\) is the heat rejected in the condenser
• \(c_p\) is the specific heat capacity of water.

Efficient cooling is essential for the condenser to maintain low-pressure steam, crucial for improving thermal efficiency. This process ensures the Rankine cycle continues to operate effectively, underscoring the importance of condenser cooling water management.