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Superheated steam at \(8 \mathrm{MPa}\) and \(480^{\circ} \mathrm{C}\) leaves the steam generator of a vapor power plant. Heat transfer and frictional effects in the line connecting the steam generator and the turbine reduce the pressure and temperature at the turbine inlet to \(7.6 \mathrm{MPa}\) and \(440^{\circ} \mathrm{C}\), respectively. The pressure at the exit of the turbine is \(10 \mathrm{kPa}\), and the turbine operates adiabatically. Liquid leaves the condenser at \(8 \mathrm{kPa}, 36^{\circ} \mathrm{C}\). The pressure is increased to \(8.6\) MPa across the pump. The turbine and pump isentropic efficiencies are \(88 \%\). The mass flow rate of steam is \(79.53 \mathrm{~kg} / \mathrm{s}\). Determine (a) the net power output, in \(\mathrm{kW}\). (b) the thermal efficiency. (c) the rate of heat transfer from the line connecting the steam generator and the turbine, in \(\mathrm{kW}\). (d) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{s}\), if the cooling water enters at \(15^{\circ} \mathrm{C}\) and exits at \(35^{\circ} \mathrm{C}\) with negligible pressure change.

Step by step solution

01

- Write down the given information

Record all given values from the problem: Superheated steam conditions at generator: Pressure = 8 MPa, Temperature = 480°C. Pressure and temperature at turbine inlet: Pressure = 7.6 MPa, Temperature = 440°C. Pressure at turbine exit: Pressure = 10 kPa. Conditions at condenser exit: Pressure = 8 kPa, Temperature = 36°C. Pressure after pump: Pressure = 8.6 MPa. Turbine and pump isentropic efficiencies = 88%. Mass flow rate of steam = 79.53 kg/s.

02

- Determine the specific enthalpy values

Use steam tables or Mollier charts to find specific enthalpies: From the steam generator: At 8 MPa and 480°C, find specific enthalpy ( \( h_{1} \)). From turbine inlet: At 7.6 MPa and 440°C, find specific enthalpy (h2). From turbine exit (isentropic process): At 10 kPa, find h2s (isentropic). Apply isentropic efficiency to find actual h3: \( h_{3} = h_{2} - \eta_{turbine} \times (h_{2} - h_{2s})\) From condenser outlet: At 8 kPa and 36°C, find specific enthalpy (h4).

03

- Find specific enthalpy after pump

Determine isentropic specific enthalpy after pump (h5s): Use the pump work formula and efficiencies. \( h_{5s} = h_{4} + \frac{v_{4}(P_5 - P_4)}{\eta_{pump}}\) Then actual enthalpy: \(h_{5} = h_{4} + \frac{v_{4}(P_5 - P_4)}{\eta_{pump}}\) Refer to the specific volume (v4) at the condenser outlet.

04

- Calculate work done by turbine and pump

Calculate work done by the turbine ( \( W_{turbine}\)): \( W_{turbine} = m_{dot} \times (h_{2} - h_{3}) \) Calculate work required for the pump (Wpump): \( W_{pump} = m_{dot} \times (h_{5} - h_{4}) \)

05

- Calculate net power output

Net power output (Wnet): \( W_{net} = W_{turbine} - W_{pump} \). Insert the calculated values into the formula.

06

- Calculate thermal efficiency

Thermal efficiency (η): \( \eta = \frac{W_{net}}{m_{dot} \times (h_{1} - h_{5})} \). Find h1 and h5 values from steps 2 and 3, and insert into the formula.

07

- Calculate heat transfer in the line

Heat transfer from the line (Qline): \( Q_{line} = m_{dot} \times (h_{1} - h_{2}) \). Find h1 and h2 values from previous steps and insert into the formula.

08

- Calculate the mass flow rate of condenser cooling water

Determine mass flow rate of condenser cooling water: Qout = \( m_{dot} \times (h_{4} - h_{6}) \) The enthalpy of water at 15°C (inlet) is h6 and 35°C (outlet) is h7. Equate: \( Q_{out} = m_{dot_{water}} \times (h_{7} - h_{6}) \).Solve for \( m_{dot_{water}} \).

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Key Concepts

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

Superheated Steam

Superheated steam is steam that is heated beyond the boiling point of water without changing its phase. This means it exists in a zone where its temperature and pressure are higher than that of saturated steam. In a vapor power plant, superheated steam is utilized to maximize the efficiency of the turbine. Such high-temperature, high-pressure steam expands through the turbine blades, generating mechanical work which is then converted into electricity.

Isentropic Efficiency

Isentropic efficiency is a measure of the actual performance of a thermodynamic process compared to an ideal, or isentropic, process. In other words, it quantifies how closely a real process, such as energy conversion in turbines and pumps, approaches the ideal isentropic process where no entropy is generated.
The efficiency can be calculated using the formula for turbines and pumps:
For Turbines: \( \text{Efficiency}_{turbine} = \frac{h_{2} - h_{3}}{h_{2} - h_{2s}} \) For Pumps: \( \text{Efficiency}_{pump} = \frac{h_{5s} - h_{4}}{h_{5} - h_{4}} \)
Here, the subscripts represent different states of the fluid (e.g., 's' represents the isentropic state, and other numbers represent different state points in the cycle). Higher isentropic efficiency indicates a process closer to the ideal process, thus less energy is lost.

Enthalpy

Enthalpy is a property of a thermodynamic system equivalent to the total heat content or total energy of the system. It's represented by the symbol 'h' and is measured in units of energy per mass, such as Joules per kilogram (J/kg). Enthalpy plays a crucial role in the calculations for vapor power plants, as it helps determine the energy changes occurring during various processes.
In the context of a vapor power plant:

• Enthalpy changes help calculate work done by the turbine.
• It is used to determine the heat added or removed in the system.
• It plays a critical role in the efficiency calculations for both the pump and the turbine.

Specific enthalpy values can be obtained from steam tables for different pressures and temperatures, and they are crucial for solving energy balance equations in thermodynamic cycles.