91Ó°ÊÓ

Steam at \(4800 \mathrm{lbf} / \mathrm{in}^{2}, 1000^{\circ} \mathrm{F}\) enters the first stage of a supercritical reheat cycle including two turbine stages. The steam exiting the first-stage turbine at \(600 \mathrm{lbf} / \mathrm{in}^{2}\) is reheated at constant pressure to \(1000^{\circ} \mathrm{F}\). Each turbine stage and the pump have an isentropic efficiency of \(85 \%\). The condenser pressure is \(1 \mathrm{lbf}^{2} .^{2}\) If the net power output of the cycle is \(100 \mathrm{MW}\), determine (a) the rate of heat transfer to the working fluid passing through the steam generator, in MW. (b) the rate of heat transfer from the working fluid passing through the condenser, in MW. (c) the cycle thermal efficiency.

Step by step solution

01

- Determine inlet and outlet states for each component

Identify the thermodynamic states at critical points in the cycle: steam generator inlet, first-stage turbine exit, reheat inlet, reheat outlet, second-stage turbine exit, and condenser exit. Use the given pressures and temperatures to find specific enthalpies and entropies.

02

- Calculate isentropic enthalpies

Using steam tables or Mollier diagram, determine the isentropic enthalpies for the turbine and pump. Remember to account for isentropic efficiency hence: \(h_{out,actual} = h_{in} - \frac{(\/h_{in} - h_{out,isentropic}}{η_{is}}\).

03

- Find turbine work output

Calculate the work done by each stage of the turbine using: \(W_{turbine} = h_{in} - h_{out,actual}\). Sum the work from both turbine stages.

04

- Determine pump work input

Calculate the work done by the pump using the enthalpy values: \(W_{pump} = h_{out,actual} - h_{in}\).

05

- Calculate net work output

The net work output is: net work = work done by turbines - work done by pump. Use this to calculate the mass flow rate from the given net power output.

06

- Determine rate of heat transfer in steam generator

Calculate the heat added in the steam generator: \(q_{add} = h_{in} - h_{out}\). Multiply by the mass flow rate to get the heat transfer rate.

07

- Calculate rate of heat transfer in condenser

Calculate the heat rejected in the condenser: \(q_{out} = h_{in} - h_{out}\). Multiply by the mass flow rate to get the heat rejection rate.

08

- Compute thermal efficiency

Thermal efficiency can be determined by the ratio: \(η = 1-\frac{q_{out}}{q_{add}}\).

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Key Concepts

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

supercritical reheat cycle

In a supercritical reheat cycle, steam is raised to a supercritical pressure and then expanded through a turbine in stages. The key aspect here is reheating the steam between the turbine stages. This involves raising the steam temperature after it has partially expanded and lost some energy in the first turbine stage. This reheating not only improves the efficiency of the cycle but also helps in reducing the moisture content of the steam, which is beneficial for the turbine blades. For our exercise, steam enters at a supercritical pressure of 4800 lbf/in² and is reheated at a constant pressure to 1000°F after the first-stage turbine.

isentropic efficiency

Isentropic efficiency is a measure of how close a real process comes to the ideal, reversible process. For turbines and pumps, it refers to the ratio of actual work delivered or consumed to the ideal work. In our problem, the isentropic efficiency of 85% means that the actual enthalpy change is 85% of the isentropic (ideal) enthalpy change. Mathematically, this is expressed as follows: For turbines: \[ h_{out,actual} = h_{in} - \frac{(h_{in} - h_{out,isentropic})}{\eta_{is}} \]For pumps:\[h_{out,actual} = h_{in} + \frac{(h_{out,isentropic} - h_{in})}{\eta_{is}} \]

thermal efficiency

Thermal efficiency is an essential metric that indicates how effectively a cycle converts heat into work. It is the ratio of net work output to the heat added in the system. In the supercritical reheat cycle, two vital rates of heat transfer need to be considered: the rate of heat transfer in the steam generator and the rate of heat transfer in the condenser. The net work is determined by subtracting the pump work from the total turbine work. The thermal efficiency can then be calculated using the formula:\[ \eta = 1-\frac{q_{out}}{q_{add}} \]where- \( q_{add} \) is the rate of heat transfer to the steam in the generator, and- \( q_{out} \) is the rate of heat transfer from the steam in the condenser.

steam turbine

A steam turbine is a key component in thermal power cycles like the one in this problem. It converts thermal energy from steam into mechanical work. The steam expands and partially cools as it moves through the turbine, causing the turbine blades to spin. In our exercise, there are two turbine stages:

• First-stage Turbine: Steam enters at 4800 lbf/in² and 1000°F, expands to 600 lbf/in².
• Second-stage Turbine: After reheating to 1000°F, it expands to the condenser pressure of 1 lbf/in².

The work produced in these stages is calculated by deriving the difference in enthalpy between the inlet and actual outlet states using the isentropic efficiency values for precision.

heat transfer

Heat transfer is a central concept in thermodynamic cycles, involving the transfer of thermal energy into and out of the working fluid. In our problem, heat is transferred in two significant processes:

• Heat Addition: Occurs in the steam generator where the heat is added to convert water into high-pressure, high-temperature steam. Calculate it using the enthalpy change from the feed pump outlet to the first-stage turbine inlet.
• Heat Rejection: Happens in the condenser where the exhaust steam from the second-stage turbine is cooled and condensed back into water. Calculate using the enthalpy change from the turbine outlet to the condenser outlet.

These heat transfers are essential for determining the cycle's thermal efficiency and ensuring the continuity of the working fluid.