91Ó°ÊÓ

Water is the working fluid in a Rankine cycle modified to include one closed feedwater heater and one open feedwater heater. Superheated vapor enters the 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 \(120 \mathrm{~kg} / \mathrm{s}\). The closed feedwater heater uses extracted steam at \(4 \mathrm{MPa}\), and the open feedwater heater uses extracted steam at \(0.3 \mathrm{MPa}\). Saturated liquid condensate drains from the closed feedwater heater at \(4 \mathrm{MPa}\) and is trapped into the open feedwater heater. The feedwater leaves the closed heater at \(16 \mathrm{MPa}\) and a temperature equal to the saturation temperature at \(4 \mathrm{MPa}\). Saturated liquid leaves the open heater at \(0.3 \mathrm{MPa}\). Assume all turbine stages and pumps operate isentropically. Determine (a) the net power developed, in kW. (b) the rate of heat transfer to the steam passing through the steam generator, 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 State Points Using Steam Tables

Utilize steam tables to find properties at different state points. Identify the enthalpies and entropies for each state: 1, 2, 3, etc. This includes steam at turbine inlet, isentropic expansion levels, feedwater heater exits, and condenser.

02

- Energy Balance for Main Turbine

Compute the work produced by the main turbine using the enthalpy change: turbine work: \(W_T = \text{mass flow rate} \times (h_{in} - h_{out})\).

03

- Energy Balance for Pumps

Determine the work required by each pump in the cycle. Use: pump work: \(W_P = \text{mass flow rate} \times (h_{out} - h_{in})\).

04

- Heat Transfer in the Steam Generator

Calculate the heat added in the boiler (steam generator) as the water is heated from the pump exit to the turbine inlet: heat added: \(Q_{in} = \text{mass flow rate} \times (h_{out} - h_{in})\).

05

- Net Power Developed

The net power is the difference between turbine work and pump work: \(W_{net} = \text{Total turbine work} - \text{Total pump work}\).

06

- Thermal Efficiency

Using the net power developed and the heat added, determine the thermal efficiency: \( \text{Efficiency} = \frac{W_{net}}{Q_{in}} \).

07

- Cooling Water Requirement for Condenser

Determine the mass flow rate of cooling water needed in the condenser by setting up an energy balance for the condenser: \( Q_{out} = \text{mass flow rate of steam} \times \text{enthalpy change of steam} = \text{mass flow rate of water} \times \text{specific heat of water} \times \text{temperature change of water} \). Solve for \( \text{mass flow rate of water} \).

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.

Thermal Efficiency

Thermal efficiency is a measure of a Rankine cycle's performance. It tells us how well the cycle converts heat energy into useful work. You can determine the thermal efficiency using the formula: \[ \eta = \frac{W_{net}}{Q_{in}} \]. Here, \(W_{net}\) is the net work output, and \(Q_{in}\) is the heat input. Start by calculating \(W_{net}\), the difference between the total turbine work and the total pump work. This value represents the useful work. Then, calculate \(Q_{in}\), the heat input to the cycle. Typically, this is the heat added in the steam generator. The ratio of these two values gives the thermal efficiency. It's essential to have accurate data for enthalpies and mass flow rates to calculate these values correctly.

Isentropic Process

An isentropic process is a reversible adiabatic process, meaning there is no heat transfer and entropy remains constant. Rankine cycles often assume isentropic conditions for turbines and pumps, simplifying calculations and closely approximating real-world performance. For example, if you need to determine the work done by a turbine, you use the initial and final enthalpies, assuming no entropy change. This allows you to apply the formula: \[ W_T = \dot{m} \times (h_{in} - h_{out}) \], where \( \dot{m} \) is the mass flow rate, \( h_{in} \) is the inlet enthalpy, and \( h_{out} \) is the outlet enthalpy. Similarly, for pumps, the work required can be determined using a similar approach. Maintaining isentropic conditions ensures the most efficient energy transfer.

Energy Balance

Understanding energy balance is crucial in the Rankine cycle. It ensures all energy inputs and outputs are accounted for. In the context of this problem, consider several key components: turbines, pumps, and the steam generator. For the turbine: \[ W_T = \dot{m} \times (h_{in} - h_{out}) \]. For the pump: \[ W_P = \dot{m} \times (h_{out} - h_{in}) \]. And for the steam generator: \[ Q_{in} = \dot{m} \times (h_{out} - h_{in}) \]. Ensure that the energy leaving each system matches the energy entering minus any work done by or on the system. This principle helps check the accuracy of your calculations and ensures all components perform correctly.

Steam Tables

Steam tables are essential tools for solving Rankine cycle problems. They provide thermal properties of water and steam at various temperatures and pressures—crucial for determining enthalpies, entropies, and other properties. For this exercise, you would use steam tables to find properties at specific state points, like the superheated vapor entering the turbine at 16 MPa and 560°C. Here, you'll look up corresponding enthalpies and entropies. These values are critical for calculating work and heat transfer. For example, use these properties to find enthalpy differences in the turbine: \[ W_T = \dot{m} \times (h_{in} - h_{out}) \]. Accurate use of steam tables ensures precise calculations and is fundamental in thermodynamic analysis.

Heat Transfer

Heat transfer is a key concept in the Rankine cycle, representing the energy exchanged between the system and its surroundings. In this cycle, heat is added in the steam generator and removed in the condenser. Calculating the heat added involves determining the enthalpy increase from the pump exit to the turbine inlet: \[ Q_{in} = \dot{m} \times (h_{out} - h_{in}) \]. Similarly, the heat removed in the condenser can be calculated using the enthalpy drop of the steam as it condenses. This is important for determining cooling water requirements: \[ Q_{out} = \dot{m_s} \times (h_{in} - h_{out}) = \dot{m_w} \times c_w \times \Delta T \]. Here, \( \dot{m_s} \) is the steam mass flow rate, \(\dot{m_w} \) is the mass flow rate of cooling water, \(c_w\) is the specific heat of water, and \(\Delta T\) is the temperature change of the cooling water. Understanding heat transfer is vital for designing efficient cooling systems and optimizing cycle performance.