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Ch 19: Work, Heat, and the First Law of Thermodynamics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 19, Problem 48

A typical nuclear reactor generates 1000 MW (1000 MJ/s) of electric energy. In doing so, it produces 2000 MW of 'waste heat' that must be removed from the reactor to keep it from melting down. Many reactors are sited next to large bodies of water so that they can use the water for cooling. Consider a reactor where the intake water is at 18°C. State regulations limit the temperature of the output water to 30°C so as not to harm aquatic organisms. How many liters of cooling water have to be pumped through the reactor each minute?

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Step 1: Understand the problem. The reactor generates 2000 MW of waste heat that needs to be removed. The cooling water absorbs this heat, and its temperature rises from 18°C to 30°C. We need to calculate the volume of water required per minute to absorb this heat without exceeding the temperature limit.
Step 2: Use the formula for heat transfer: \( Q = m \cdot c \cdot \Delta T \), where \( Q \) is the heat energy, \( m \) is the mass of water, \( c \) is the specific heat capacity of water (approximately \( 4.186 \; \text{J/g°C} \)), and \( \Delta T \) is the temperature change (\( 30°C - 18°C = 12°C \)).
Step 3: Convert the waste heat \( Q \) from megawatts to joules per minute. Since \( 1 \; \text{MW} = 10^6 \; \text{J/s} \), the waste heat is \( 2000 \; \text{MW} \cdot 60 \; \text{s/min} = 1.2 \times 10^{11} \; \text{J/min} \).
Step 4: Rearrange the heat transfer formula to solve for \( m \), the mass of water: \( m = \frac{Q}{c \cdot \Delta T} \). Substitute \( Q = 1.2 \times 10^{11} \; \text{J/min} \), \( c = 4.186 \; \text{J/g°C} \), and \( \Delta T = 12°C \) into the equation.
Step 5: Convert the mass of water \( m \) from grams to liters. Since the density of water is approximately \( 1 \; \text{g/cm}^3 \), \( 1 \; \text{g} \) of water corresponds to \( 1 \; \text{cm}^3 \), or \( 0.001 \; \text{liters} \). Divide the mass \( m \) by 1000 to find the volume of water in liters per minute.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one. In the context of a nuclear reactor, waste heat generated during energy production must be effectively removed to prevent overheating. This can occur through conduction, convection, or radiation, with convection being the primary method when using water as a coolant.
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Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is approximately 4.18 J/g°C. Understanding specific heat capacity is crucial for calculating how much water is needed to absorb the waste heat produced by the reactor while keeping the output temperature within regulatory limits.
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Flow Rate

Flow rate refers to the volume of fluid that passes through a given surface per unit time, typically measured in liters per minute or cubic meters per second. In this scenario, calculating the required flow rate of cooling water involves determining how much water must be circulated to absorb the waste heat while maintaining the temperature constraints set by regulations.
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Related Practice
Textbook Question

A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is 20°C, the piston floats 20 cm above the bottom of the cylinder. What is the gas pressure?

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Textbook Question

2.0 mol of gas are at 30 °C and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at constant pressure?

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Textbook Question

512 g of an unknown metal at a temperature of 15°C is dropped into a 100 g aluminum container holding 325 g of water at 98°C. A short time later, the container of water and metal stabilizes at a new temperature of 78°C. Identify the metal.

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Textbook Question

The beaker in FIGURE P19.45, with a thin metal bottom, is filled with 20 g of water at 20°C. It is brought into good thermal contact with a 4000 cm3 container holding 0.40 mol of a monatomic gas at 10 atm pressure. Both containers are well insulated from their surroundings. What is the gas pressure after a long time has elapsed? You can assume that the containers themselves are nearly massless and do not affect the outcome.

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Textbook Question

When air is inhaled, it quickly becomes saturated with water vapor as it passes through the moist airways. Consequently, an adult human exhales about 25 mg of evaporated water with each breath. Evaporation—a phase change—requires heat, and the heat energy is removed from your body. Evaporation is much like boiling, only water's heat of vaporization at 35°C is a somewhat larger 24×105 J/kg because at lower temperatures more energy is required to break the molecular bonds. At 12 breaths/min, on a dry day when the inhaled air has almost no water content, what is the body's rate of energy loss (in J/s) due to exhaled water? (For comparison, the energy loss from radiation, usually the largest loss on a cool day, is about 100 J/s.)

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Textbook Question

A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is 20°C, the piston floats 20 cm above the bottom of the cylinder. What is the new equilibrium temperature of the gas?

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