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A combined gas turbine-vapor power plant (Fig. 9.23) has a net power output of \(100 \mathrm{MW}\). Air enters the compressor of the gas turbine at \(100 \mathrm{kPa}, 300 \mathrm{~K}\), and is compressed to \(1200 \mathrm{kPa}\). The isentropic efficiency of the compressor is \(84 \%\). The conditions at the inlet to the turbine are \(1200 \mathrm{kPa}\) and \(1400 \mathrm{~K}\). Air expands through the turbine, which has an isentropic efficiency of \(88 \%\), to a pressure of \(100 \mathrm{kPa}\). The air then passes through the interconnecting heat exchanger, and is finally discharged at \(480 \mathrm{~K}\). Steam enters the turbine of the vapor power cycle at \(8 \mathrm{MPa}, 400^{\circ} \mathrm{C}\), and expands to the condenser pressure of \(8 \mathrm{kPa}\). Water enters the pump as saturated liquid at \(8 \mathrm{kPa}\). The turbine and pump have isentropic efficiencies of 90 and \(80 \%\), respectively. Determine (a) the mass flow rates of air and steam, each in \(\mathrm{kg} / \mathrm{s}\). (b) the thermal efficiency of the combined cycle. (c) a full accounting of the net exergy increase of the air passing through the combustor of the gas turbine, \(\dot{m}_{\text {air }}\left[\mathrm{e}_{\mathrm{f} 3}-\mathrm{e}_{\mathrm{f} 2}\right]\), in MW. Discuss. Let \(T_{0}=300 \mathrm{~K}, p_{0}=100 \mathrm{kPa}\).

Step by step solution

01

- Find Properties at Various States

Determine the properties such as enthalpy and entropy at the various states that air and steam pass through in the respective cycles using thermodynamic tables or software.

02

- Gas Turbine Cycle: Compressor Work

Calculate the work input to the compressor using the isentropic efficiency. The isentropic compressor work \(W_{c,s}\) can be found using the relation \[ W_{c,s} = c_p (T_{2s} - T_1) \] where \(T_{2s}\) is the temperature obtained from isentropic relations. The actual compressor work \(W_c\) can be calculated as: \[ W_c = \frac{W_{c,s}}{\eta_c} \] where \(\eta_c = 0.84\).

03

- Gas Turbine Cycle: Turbine Work

Calculate the work output of the turbine using the isentropic efficiency. The isentropic turbine work \(W_{t,s}\) can be found using the relation \[ W_{t,s} = c_p (T_3 - T_{4s}) \] where \(T_{4s}\) is the temperature obtained from isentropic relations. The actual turbine work \(W_t\) can be calculated as: \[ W_t = W_{t,s} \cdot \eta_t \] where \(\eta_t = 0.88\).

04

- Gas Turbine Cycle: Heat Addition

Calculate the heat added in the gas turbine cycle using the relation \[ Q_{in} = c_p (T_3 - T_2) \]

05

- Vapor Power Cycle: Pump Work

Calculate the pump work using the isentropic efficiency. First, find the isentropic pump work \[ W_{p,s} = v (p_2 - p_1) \] The actual pump work \(W_p\) can be calculated as: \[ W_p = \frac{W_{p,s}}{\eta_p} \] where \(\eta_p = 0.8\) and \(v\) is the specific volume of the liquid at the pump inlet.

06

- Vapor Power Cycle: Turbine Work

Calculate the turbine work output for the vapor power cycle. First, find the isentropic turbine work \[ W_{t,v,s} = h_3 - h_{4s} \] The actual turbine work \(W_{t,v}\) can be calculated as: \[ W_{t,v} = W_{t,v,s} \cdot \eta_{t,v} \] where \(\eta_{t,v} = 0.9\).

07

- Vapor Power Cycle: Heat Addition

Calculate the heat addition in the vapor power cycle using the enthalpies: \[ Q_{in,v} = h_3 - h_2 \]

08

- Mass Flow Rates

Using the power output requirement and the work interactions, determine the mass flow rates of both air and steam. The mass flow rate of air can be found from \[ m_{air} = \frac{Net Power}{W_t - W_c} \] The mass flow rate of steam can be found using \[ m_{steam} = \frac{Net Power}{W_{t,v} - W_p} \]

09

- Thermal Efficiency of Combined Cycle

Calculate the thermal efficiency (\(\eta_{thermal}\)) of the combined cycle using the formula: \[ \eta_{thermal} = \frac{Net Power Output}{Q_{in} + Q_{in,v}} \]

10

- Exergy Increase through Combustor

Calculate the net exergy increase of the air passing through the combustor using the exergy change formula: \[ \dot{m}_{air}[e_{f3} - e_{f2}] \] where \(e_f\) is the specific flow exergy.

11

- Discussion

Discuss the significance of the net exergy increase and the entropy generation through the process.

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Key Concepts

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

Understanding Mass Flow Rate Calculation

In a gas turbine-vapor power plant, mass flow rate is crucial because it determines how much working fluid (air or steam) is moving through the system. The mass flow rate of air can be calculated using the net power output and the work done by the compressor and turbine. The formula is:
\( m_{air} = \frac{Net Power}{W_t - W_c} \)
Here, \(W_t\) represents turbine work and \(W_c\) is compressor work. Similarly, the mass flow rate of steam is found from:
\( m_{steam} = \frac{Net Power}{W_{t,v} - W_p} \)
where \(W_{t,v}\) is the turbine work in the vapor cycle, and \(W_p\) represents the pump work. Calculating these accurately ensures the plant operates efficiently.

Calculating Thermal Efficiency

Thermal efficiency (\(\eta_{thermal}\)) of a combined cycle power plant measures how well it converts heat into work or electrical energy. It's given by:
\( \eta_{thermal} = \frac{Net Power Output}{Q_{in} + Q_{in,v}} \)
Here, \(Net Power Output\) is the power generated, \(Q_{in}\) represents the heat added to the gas turbine cycle, and \(Q_{in,v}\) is the heat added to the vapor cycle. Higher thermal efficiency means more effective utilization of the heat energy supplied to the plant.

Isentropic Efficiency and Its Importance

Isentropic efficiency compares the actual performance of compressors and turbines to ideal, reversible processes. For a compressor, it's:
\( \eta_c = \frac{W_{c,s}}{W_c} \)
where \(W_{c,s}\) is isentropic compressor work and \(W_c\) is actual work. Similarly, for a turbine:
\( \eta_t = \frac{W_t}{W_{t,s}} \)
where \(W_t\) is the actual turbine work and \(W_{t,s}\) is isentropic work. Higher isentropic efficiency indicates better performance with less energy loss.

Conducting Exergy Analysis

Exergy analysis helps us understand the useful work potential of a system. For air moving through the combustor, we consider the net exergy increase:
\( \dot{m}_{air}[e_{f3} - e_{f2}] \)
Here, \(\dot{m}_{air}\) is the mass flow rate, and \(e_f\) represents specific flow exergy. The difference \(e_{f3} - e_{f2}\) indicates how much useful work has been added. This reveals how effectively the combustion process is converting fuel energy into work.

Evaluating Heat Addition

Heat addition in the cycles is crucial for energy conversion. In the gas turbine cycle:
\( Q_{in} = c_p (T_3 - T_2) \)
Here, \(c_p\) is the specific heat of air, and \(T_3\) and \(T_2\) are temperatures before and after heat addition. In the vapor cycle:
\( Q_{in,v} = h_3 - h_2 \)
where \(h_3\) and \(h_2\) are the enthalpies. Proper heat addition ensures maximum work extraction from the thermal energy provided.

Understanding Compressor Work

Compressor work significantly affects the efficiency of the gas turbine cycle. You calculate the isentropic work first using:
\( W_{c,s} = c_p (T_{2s} - T_1) \)
Then, using isentropic efficiency:
\( W_c = \frac{W_{c,s}}{\eta_c} \)
Accurate calculation ensures the compressor is adding necessary energy to pressurize the air without excessive energy loss.

Calculating Turbine Work

Turbine work is about extracting usable energy. For the gas turbine cycle:
\( W_{t,s} = c_p (T_3 - T_{4s}) \)
Then use isentropic efficiency to find actual work:
\( W_t = W_{t,s} \cdot \eta_t \)
For the vapor cycle:
\( W_{t,v,s} = h_3 - h_{4s} \)
and
\( W_{t,v} = W_{t,v,s} \cdot \eta_{t,v} \)
Correctly calculating turbine work ensures the cycle generates maximum power.