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A nuclear-fueled electric power plant utilizes a so-called "boiling water reactor." In this type of reactor, nuclear energy causes water under pressure to boil at \(285^{\circ} \mathrm{C}\) (the temperature of the hot reservoir). After the steam does the work of turning the turbine of an electric generator, the steam is converted back into water in a condenser at \(40^{\circ} \mathrm{C}\) (the temperature of the cold reservoir). To keep the condenser at \(40^{\circ} \mathrm{C},\) the rejected heat must be carried away by some means \(-\) for example, by water from a river. The plant operates at three-fourths of its Carnot efficiency, and the electrical output power of the plant is \(1.2 \times 10^{9}\) watts. A river with a water flow rate of \(1.0 \times 10^{5} \mathrm{kg} / \mathrm{s}\) is available to remove the rejected heat from the plant. Find the number of Celsius degrees by which the temperature of the river rises.

Step by step solution

01

Calculate Carnot Efficiency

The Carnot efficiency (\(\eta_{\text{carnot}}\)) is determined using the formula:\[\eta_{\text{carnot}} = 1 - \frac{T_C}{T_H}\]where \(T_C\) and \(T_H\) are the cold and hot reservoir temperatures in Kelvin. Convert the Celsius temperatures: - Hot reservoir: \(T_H = 285 + 273.15 = 558.15\, \text{K}\) - Cold reservoir: \(T_C = 40 + 273.15 = 313.15\, \text{K}\)Then, plug these into the Carnot efficiency formula:\[\eta_{\text{carnot}} = 1 - \frac{313.15}{558.15} \approx 0.4386\] (or \(43.86\%\))

02

Adjust for Actual Efficiency

The plant operates at three-fourths of its Carnot efficiency, so calculate the actual efficiency (\(\eta_{\text{actual}}\)) using:\[\eta_{\text{actual}} = \frac{3}{4} \times \eta_{\text{carnot}} = \frac{3}{4} \times 0.4386 = 0.32895\] (or \(32.895\%\))

03

Determine Heat Input

The heat input rate (\(Q_H\)) can be calculated from the electrical output power (\(P = 1.2 \times 10^9\) watts) using:\[P = \eta_{\text{actual}} \times Q_H\]Solving for \(Q_H\):\[Q_H = \frac{P}{\eta_{\text{actual}}} = \frac{1.2 \times 10^9}{0.32895} \approx 3.648 \times 10^9\, \text{watts}\]

04

Calculate Rejected Heat

The rejected heat rate (\(Q_C\)) can be found by the difference between the heat input and output powers:\[Q_C = Q_H - P = 3.648 \times 10^9 - 1.2 \times 10^9 = 2.448 \times 10^9\, \text{watts}\]

05

Calculate Temperature Rise in River

The temperature rise in the river water (\(\Delta T\)) is determined by:\[Q_C = m \cdot c \cdot \Delta T\]where: - \(m = 1.0 \times 10^5\, \text{kg/s}\) (mass flow rate of river)- \(c = 4186\, \text{J/kg}\cdot\text{K}\) (specific heat capacity of water)Solving for \(\Delta T\):\[\Delta T = \frac{Q_C}{m \cdot c} = \frac{2.448 \times 10^9}{1.0 \times 10^5 \times 4186} \approx 5.85\, ^{\circ}\text{C}\]

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Key Concepts

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

Nuclear Power Plant

A nuclear power plant is a facility that generates electricity by using nuclear reactions as a source of heat. The heat is used to produce steam, which drives a steam turbine connected to an electric generator. These plants rely on the principle of nuclear fission, where the nucleus of an atom splits into smaller parts, releasing a substantial amount of energy.

Nuclear power plants are known for their potential to generate large amounts of electricity with a relatively low environmental footprint compared to fossil fuel-based plants. Yet, there are concerns about radioactive waste and the potential for severe accidents.

• They operate continuously, providing a consistent power supply.
• They play a crucial role in energy independence and security.
• They have a high capital cost but low operational expenses.

Boiling Water Reactor

A boiling water reactor (BWR) is a type of nuclear reactor used in nuclear power plants where water is used as both a coolant and a neutron moderator. In a BWR, water boils inside the reactor core to directly generate steam. This steam is then used to drive a turbine that produces electricity.

BWRs are simpler in design compared to other reactor types like pressurized water reactors (PWRs), as they do not require a separate heat exchanger to generate steam. This simplicity can lead to cost savings and efficiency.

However, BWRs pose challenges related to steam quality and require careful control to avoid releasing radioactive material. The steam and water that pass through the reactor also run through the turbine, exposing it to radiation.

• They feature a direct cycle where water is boiled in the core.
• They require rigorous safety measures due to radioactive steam.
• They generally have a good operational efficiency.

Heat Transfer

Heat transfer is a fundamental concept in understanding how a nuclear power plant operates, particularly in the context of energy conversion processes. In the plant, heat produced during nuclear fission is transferred to water, turning it into steam.

There are three modes of heat transfer: conduction, convection, and radiation. In a nuclear plant, conduction and convection are most relevant. Conduction occurs when heat is transferred through solid materials like fuel rods, while convection involves the movement of heat through fluids such as water or steam.

The efficiency of heat transfer affects the overall efficiency of the power plant. Less efficient heat transfer means more heat is rejected, typically to a nearby river or cooling tower. Proper management of heat transfer is vital to maintain plant safety and operational performance.

• It is essential for converting nuclear energy into electricity efficiently.
• Involves both conduction within materials and convection in fluids.
• Impacts the cooling systems and environmental heat dissipation.

Steam Turbine

The steam turbine is a key component in a nuclear power plant, converting thermal energy from steam into mechanical energy to produce electricity. In a boiling water reactor plant, steam generated from boiling water directly drives the turbine.

Steam turbines operate on the principles of thermodynamics, using high-pressure steam to spin a rotor. This spinning motion is converted into electrical energy by a generator.

The efficiency and performance of the steam turbine are crucial to the power plant's overall effectiveness. Factors such as steam pressure, temperature, and turbine blade design can impact performance.

Steam turbines must be carefully monitored for maintenance and efficiency improvements.

• They transform steam's thermal energy into rotational mechanical energy.
• They need precise control of steam conditions for optimal performance.
• They are central to energy conversion in nuclear power plants.