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Water is the working fluid in an ideal regenerative Rankine cycle with one open feedwater heater. Steam enters the turbine at 1400 lbf/in. \({ }^{2}\) and \(1000^{\circ} \mathrm{F}\) and expands to \(120 \mathrm{lbf} / \mathrm{in} .^{2}\), where some of the steam is extracted and diverted to the open feedwater heater operating at \(120 \mathrm{lbf}^{2}\). \(^{2}\) The remaining steam expands through the second-stage turbine to the condenser pressure of 2 lbf/in. \({ }^{2}\) Saturated liquid exits the open feedwater heater at \(120 \mathrm{lbf}^{2} .^{2}\) The net power output of the cycle is \(1 \times 10^{9} \mathrm{~B}\) tu/h. Determine for the cycle (a) the mass flow rate of steam entering the first stage of the turbine, in \(\mathrm{lb} / \mathrm{h}\). (b) the rate of heat transfer, in Btu/h, to the working fluid passing through the steam generator. (c) the thermal efficiency.

Step by step solution

01

Identify Given Data

Extract the data given in the problem: - Steam enters the turbine at 1400 lbf/in. - Temperature is 1000°F.- Pressure after expansion: 120 lbf/in^2.- Condenser pressure: 2 lbf/in^2.- Saturated liquid exits the feedwater heater at 120 lbf/in^2.- Net Power Output: \(1 \times 10^9 \text{ Btu/h}\). Define the states involved: State 1: Before first turbine expansion, State 2: After first turbine expansion, State 3: Before second turbine expansion, State 4: After second turbine expansion.

02

Determine Thermodynamic Properties

For each state, determine enthalpy (h) and entropy (s) using steam tables or Mollier chart.State 1: \( P_1 = 1400 \text{ lbf/in}^2\), \( T_1 = 1000^\text{°}F\).State 2: \( P_2 = 120 \text{ lbf/in}^2\) (after partial expansion in the turbine).State 3: Extracted steam and feedwater heater operating pressure: \( P_3 = 120 \text{ lbf/in}^2\).State 4: Remaining steam expands to condenser pressure: \( P_4 = 2 \text{ lbf/in}^2\).Look up and record the corresponding enthalpies.

03

Apply Energy Balance for Turbines and Heater

Using the energy balance equation for the first turbine stage: \[ h_2 = h_1 - (h_1 - h_2) \]\ For the open feedwater heater, applying mass and energy balance gives: \[ m_1h_3 + m_2h_2 = m_3h_f + m_fh_3 \]\ where \(m\) is mass flow rate.

04

Solve for Mass Flow Rates

Using the net power output and the enthalpy values determined, calculate the mass flow rate. Using turbine work formula: \[ W_{\text{net}} = \frac{\text{Net Output}}{\text{Specific Enthalpy Change}}\] Find mass flow rates entering the first turbine and feedwater heater.

05

Determine Heat Transfer Rate

Use energy balance for the steam generator: \[ Q_{\text{in}} = m \times (h_1 - h_f)\]\ where \(h_f\) is the enthalpy at the feedwater heater exit state.

06

Calculate Thermal Efficiency

Thermal Efficiency can be found using the formula: \[ \text{Efficiency}(\text{η}) = \frac{W_{\text{net}}}{Q_{\text{in}}} \]

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Key Concepts

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

Thermal Efficiency

Thermal efficiency is a measure of how well a Rankine cycle converts heat into work. In a regenerative Rankine cycle, some steam is diverted to a feedwater heater, improving efficiency by preheating the feedwater entering the boiler. Preheating the feedwater reduces the heat added in the steam generator, thereby increasing the cycle's efficiency. To calculate thermal efficiency, we use the formula:
\ \text{Thermal Efficiency}(\text{η}) = \frac{W_{\text{net}}}{Q_{\text{in}}} \
Here, \(W_{\text{net}}\) is the net work output and \(Q_{\text{in}}\) is the heat input to the cycle. This ratio represents how much of the heat input is converted to useful work. In practical terms, a higher thermal efficiency indicates a more efficient cycle, meaning fewer energy losses and better performance.

Mass Flow Rate

Mass flow rate is the amount of working fluid (in this case, steam) passing through the system per unit time, typically expressed as \( \text{lb}/\text{h} \) or \( \text{kg}/\text{s} \). To determine the mass flow rate, we use energy balance equations and the given net power output.
For example, if the net power output is known and we have the enthalpy changes across turbines, we can calculate the mass flow rate as follows:
\ \dot{m} = \frac{W_{\text{net}}}{\Delta h} \
Here, \( \dot{m} \) is the mass flow rate, \( W_{\text{net}} \) is the net power output, and \( \Delta h \) is the specific enthalpy change. This equation shows that the mass flow rate is directly proportional to the net power output and inversely proportional to the enthalpy change across the turbine.

Heat Transfer Rate

The heat transfer rate or \( Q_{\text{in}} \), determines the amount of thermal energy added to the working fluid in the steam generator. For a regenerative Rankine cycle, this can be calculated using the following equation:
\ Q_{\text{in}} = \dot{m} \times (h_1 - h_f) \
Here, \( \dot{m} \) is the mass flow rate, \( h_1 \) is the enthalpy of the steam entering the turbine, and \( h_f \) is the enthalpy of the feedwater exiting the feedwater heater. Understanding this concept helps in optimizing the system to provide the maximum work output with minimum heat input, ultimately leading to a more efficient cycle.

Energy Balance

Energy balance is a crucial concept in thermodynamics that ensures the energy entering a system equals the energy exiting it plus any changes in internal energy. For the regenerative Rankine cycle, energy balances are applied to different components like turbines and feedwater heaters.
In a turbine, the energy balance can be represented as follows:
\ h_2 = h_1 - (h_1 - h_2) \
For the open feedwater heater, the following energy balance equation applies:
\ m_1h_3 + m_2h_2 = m_3h_f + m_fh_3 \
Here, \( h \), \( m \), denote enthalpy and mass flow rate, respectively. These balance equations ensure that all energy interactions within the system are accounted for, enabling accurate calculations.

Turbine Expansion

Turbine expansion in the regenerative Rankine cycle involves the steam expanding through one or more stages of turbines to produce work. Initially, the steam enters the first turbine stage and expands to a lower pressure, dropping its enthalpy and producing work. Some steam is diverted for regeneration (to the feedwater heater), while the remaining steam continues to the second stage of turbine expansion until it reaches the condenser pressure.

Understanding turbine expansion involves determining the enthalpy changes at each stage. The specific enthalpy change across the turbine stages helps in calculating the work produced and the enthalpy states required for further calculations. This step is crucial for analyzing the performance of the cycle and optimizing the work output.