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Steam enters the turbine of a steam power plant that operates on a simple ideal Rankine cycle at a pressure of \(6 \mathrm{MPa},\) and it leaves as a saturated vapor at \(7.5 \mathrm{kPa}\). Heat is transferred to the steam in the boiler at a rate of \(40,000 \mathrm{kJ} / \mathrm{s}\) Steam is cooled in the condenser by the cooling water from a nearby river, which enters the condenser at \(15^{\circ} \mathrm{C}\). Show the cycle on a \(T-s\) diagram with respect to saturation lines, and determine \((a)\) the turbine inlet temperature, \((b)\) the net power output and thermal efficiency, and \((c)\) the minimum mass flow rate of the cooling water required.

Short Answer

Expert verified
Answer: The turbine inlet temperature is 275.6℃.

Step by step solution

01

Analyze the given information

We are given the cycle operates on a simple ideal Rankine cycle with steam entering the turbine at a pressure of \(6 \mathrm{MPa}\) and exiting as a saturated vapor at \(7.5 \mathrm{kPa}\).
02

Draw a T-s diagram

Draw a \(T-s\) diagram with respect to saturation lines. The process between points 1 and 2 represents the work done by steam in the turbine. The process between points 2 and 3 represents the heat transfer that occurs in the condenser. The process between points 3 and 4 represents the work done in the pump. The process between points 4 and 1 represents the heat transferred in the boiler.
03

Find the turbine inlet temperature

At the steam entry point to the turbine (point 1), the pressure is given and we need to find the corresponding saturation temperature. Using steam tables, we can determine the saturation temperature at a pressure of \(6 \mathrm{MPa}\). The turbine inlet temperature, \(T_1 = 275.6 ^\circ \mathrm{C}\)
04

Determine the enthalpy values at different points in the cycle

Using the steam tables, we can determine the enthalpy values at different points in the cycle. \(h_1 = h_\mathrm{vapor, 6MPa} = 2800.6 \frac{kJ}{kg}\) \(h_2 = h_\mathrm{vapor, 7.5kPa} = 2465.3 \frac{kJ}{kg}\) \(h_3 = h_\mathrm{liquid, 7.5kPa} = 170.75 \frac{kJ}{kg}\) \(h_4 = h_3 + v_3 \cdot (P_4 - P_3) \cdot 10^3 = 170.75 \frac{kJ}{kg} + 0.001004 \frac{m^3}{kg} \cdot (6000-7.5)kPa = 175.46 \frac{kJ}{kg}\)
05

Calculate the heat transfer rates in the boiler and the condenser

The heat transferred in the boiler, \(q_\mathrm{in} = h_1 - h_4 = 2800.6 - 175.46 = 2625.14 \frac{kJ}{kg}\) The heat transferred in the condenser, \(q_\mathrm{out} = h_2 - h_3 = 2465.3 - 170.75 = 2294.55 \frac{kJ}{kg}\)
06

Determine the net power output and mass flow rate of steam

The net power output is given by the product of mass flow rate and the difference in enthalpy across the turbine. We are given the heat transfer rate in the boiler, \(Q_\mathrm{in} = 40,000 \frac{kJ}{s}\). We can calculate the mass flow rate of steam as: \(\dot{m}_\mathrm{steam} = \frac{Q_\mathrm{in}}{q_\mathrm{in}} = \frac{40,000}{2625.14} = 15.24 \frac{kg}{s}\) Now, we can determine the net power output as: \(W_\mathrm{net} = \dot{m}_\mathrm{steam} \cdot (h_1 - h_2) = 15.24 \cdot (2800.6 - 2465.3) = 5113.57 \mathrm{kW}\)
07

Calculate the thermal efficiency of the cycle

The thermal efficiency is given by the ratio of net power output to the heat input in the boiler: \(\eta_\mathrm{thermal} = \frac{W_\mathrm{net}}{Q_\mathrm{in}} = \frac{5113.57}{40,000} = 0.1278 = 12.78\%\)
08

Determine the minimum mass flow rate of cooling water

We can find the minimum mass flow rate of cooling water by considering the heat transfer in the condenser. The heat transfer rate to the cooling water is given by: \(Q_\mathrm{out} = \dot{m}_\mathrm{cw} \cdot c_\mathrm{p} \cdot (T_\mathrm{out} - T_\mathrm{in})\) Where \(\dot{m}_\mathrm{cw}\) is the mass flow rate of cooling water, \(c_\mathrm{p} = 4.18 \frac{kJ}{kg \cdot K}\) is the specific heat capacity of water, \(T_\mathrm{out}\) is the cooling water outlet temperature, and \(T_\mathrm{in} = 15^{\circ}\mathrm{C}\) is the cooling water inlet temperature. From the energy balance, we know the heat transfer rate out of the cycle is given by the mass flow rate of steam and the heat transferred in the condenser: \(Q_\mathrm{out} = \dot{m}_\mathrm{steam} \cdot q_\mathrm{out} = 15.24 \cdot 2294.55 = 34,951.3 \frac{kJ}{s}\) Assuming the cooling water outlet temperature is the saturation temperature at the condenser pressure \(7.5 \mathrm{kPa}\), we find \(T_\mathrm{out} = 30.1^{\circ}\mathrm{C}\) Now we can find the mass flow rate of cooling water: \(\dot{m}_\mathrm{cw} = \frac{Q_\mathrm{out}}{c_\mathrm{p} \cdot (T_\mathrm{out} - T_\mathrm{in})} = \frac{34,951.3}{4.18 \cdot (30.1 - 15)} = 668 \frac{kg}{s}\)
09

Summary of Results:

(a) Turbine inlet temperature: \(T_1 = 275.6^{\circ} \mathrm{C}\) (b) Net power output: \(W_\mathrm{net} = 5113.57 \mathrm{kW}\), and thermal efficiency: \(\eta_\mathrm{thermal} = 12.78\%\) (c) Minimum mass flow rate of cooling water: \(\dot{m}_\mathrm{cw} = 668 \frac{kg}{s}\)

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Key Concepts

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

T-s Diagram
The temperature-entropy (T-s) diagram is a valuable tool for visualizing and analyzing thermal cycles such as the Rankine cycle used in power plants. On the T-s diagram, the vertical axis represents temperature, while entropy is depicted on the horizontal axis.

For a Rankine cycle, the diagram helps to illustrate the various stages including the isentropic expansion in the turbine, the isobaric heat rejection in the condenser, the isentropic compression in the pump, and isobaric heat addition in the boiler. By plotting these processes, we can identify areas where energy efficiency can be improved. For instance, a wide gap between the saturation lines, known as the 'moisture dome', can give insights into the wetness of steam and thus the potential for turbine blade erosion. The T-s diagram enhances comprehension of the thermodynamics governing steam cycles and underlines the balance between entropy generation and thermal efficiency.
Thermal Efficiency
Thermal efficiency is a measure of a system's ability to convert heat into work. In the context of the Rankine cycle, it is the ratio of the net work output of the cycle to the heat input into the boiler.

The formula for thermal efficiency (\( u \text{ or sometimes } u_{\text{th}} \text{ in scientific texts}\)) can be mathematically represented as \( u = \frac{W_{\text{net}}}{Q_{\text{in}}} \), where \(W_{\text{net}}\) is the net work done by the system (which is the difference between the turbine work output and the work input to the pump) and \(Q_{\text{in}}\) is the heat supplied to the boiler.

Maximizing Efficiency

To enhance thermal efficiency, one might consider superheating the steam before it enters the turbine or increasing the temperature and pressure at which heat is supplied. However, the student should note that real-world factors like material limitations and economic considerations may constrain the achievable efficiency.
Mass Flow Rate
Mass flow rate (\( \textstyle \frac{kg}{s} \) or \( \textstyle \frac{lb}{s} \) in Imperial units) is a critical aspect of the Rankine cycle, representing the amount of mass passing through a component (like a turbine or condenser) per unit time. It is an essential parameter for calculating the energy carried by the working fluid, which in turn affects the power output and cooling requirements of the cycle.

By knowing the mass flow rate and the energy content (enthalpy) of the steam, we can determine the heat transfer rates, power output, and the efficiency of the plant. A correct estimation of the mass flow rate is vital for designing a power plant that meets the required electricity demand while ensuring the thermal load is within the limits to prevent equipment damage.

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Most popular questions from this chapter

Steam enters the high-pressure turbine of a steam power plant that operates on the ideal reheat Rankine cycle at 800 psia and \(900^{\circ} \mathrm{F}\) and leaves as saturated vapor. Steam is then reheated to \(800^{\circ} \mathrm{F}\) before it expands to a pressure of 1 psia. Heat is transferred to the steam in the boiler at a rate of \(6 \times 10^{4}\) Btu/s. Steam is cooled in the condenser by the cooling water from a nearby river, which enters the condenser at \(45^{\circ} \mathrm{F}\). Show the cycle on a \(T\) -s diagram with respect to saturation lines, and determine ( \(a\) ) the pressure at which reheating takes place, \((b)\) the net power output and thermal efficiency, and \((c)\) the minimum mass flow rate of the cooling water required.

Why is mercury a suitable working fluid for the topping portion of a binary vapor cycle but not for the bottoming cycle?

The gas-turbine cycle of a combined gas-steam power plant has a pressure ratio of \(12 .\) Air enters the compressor at \(310 \mathrm{K}\) and the turbine at \(1400 \mathrm{K}\). The combustion gases leaving the gas turbine are used to heat the steam at \(12.5 \mathrm{MPa}\) to \(500^{\circ} \mathrm{C}\) in a heat exchanger. The combustion gases leave the heat exchanger at \(247^{\circ} \mathrm{C}\). Steam expands in a high-pressure turbine to a pressure of \(2.5 \mathrm{MPa}\) and is reheated in the combustion chamber to \(550^{\circ} \mathrm{C}\) before it expands in a low-pressure turbine to \(10 \mathrm{kPa} .\) The mass flow rate of steam is \(12 \mathrm{kg} / \mathrm{s}\). Assuming all the compression and expansion processes to be isentropic, determine (a) the mass flow rate of air in the gas-turbine cycle, ( \(b\) ) the rate of total heat input, and ( \(c\) ) the thermal efficiency of the combined cycle.

Several geothermal power plants are in operation in the United States and more are being built since the heat source of a geothermal plant is hot geothermal water, which is "free energy." An 8 -MW geothermal power plant is being considered at a location where geothermal water at \(160^{\circ} \mathrm{C}\) is available. Geothermal water is to serve as the heat source for a closed Rankine power cycle with refrigerant-134a as the working fluid. Specify suitable temperatures and pressures for the cycle, and determine the thermal efficiency of the cycle. Justify your selections.

Using EES (or other) software, investigate the effect of the condenser pressure on the performance of a simple ideal Rankine cycle. Turbine inlet conditions of steam are maintained constant at \(10 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) while the condenser pressure is varied from 5 to 100 kPa. Determine the thermal efficiency of the cycle and plot it against the condenser pressure, and discuss the results.

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