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Chapter 16: Problem 1

A coal-fired power plant that operates at an efficiency of \(38 \%\) generates \(750 \mathrm{MW}\) of electric power. How much heat does the plant discharge to the environment in one day?

Short Answer

Expert verified
The plant discharges 29,368.32 MWh of heat to the environment in one day.

Step by step solution

01

Identify Given Values

The problem states that the power plant generates electric power at an efficiency of 38%. This is denoted as \( \eta = 0.38 \). The electric power output is given as \( P_{\text{output}} = 750 \text{ MW} \).
02

Calculate Heat Input

Using the efficiency formula \( \eta = \frac{P_{\text{output}}}{P_{\text{input}}} \), we solve for the total heat input \( P_{\text{input}} \):\[P_{\text{input}} = \frac{P_{\text{output}}}{\eta} = \frac{750}{0.38} = 1973.68 \text{ MW}\]
03

Determine Heat Discharged to the Environment

The heat discharged to the environment \( Q_{\text{discharged}} \) is the difference between the heat input and the electric power output:\[Q_{\text{discharged}} = P_{\text{input}} - P_{\text{output}} = 1973.68 \text{ MW} - 750 \text{ MW} = 1223.68 \text{ MW}\]This is the rate of heat discharged.
04

Convert Power Rate to Daily Energy

Since the problem asks for the total heat discharged over one day, we need to convert this power rate to energy in megawatt-hours (MWh). There are 24 hours in a day, so:\[Q_{ ext{daily}} = Q_{\text{discharged}} \times 24 = 1223.68 \times 24 = 29368.32 \text{ MWh}\]

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Key Concepts

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

Efficiency Calculations
Efficiency in the context of power plants is a measure of how well a plant converts fuel into usable electricity. It tells us what fraction of the fuel's energy is actually used to produce electricity and not wasted. The efficiency (\(\eta\)) is calculated using the formula:
  • \(\eta = \frac{P_{\text{output}}}{P_{\text{input}}}\)
Where:
  • \(P_{\text{output}}\) is the power produced (in MW or similar units).
  • \(P_{\text{input}}\) is the total energy input required to generate that power.
The higher the efficiency, the more of the input energy is turned into useful electricity. In the provided exercise, the efficiency of the power plant is 38%, meaning 38% of the input energy becomes electric power, while the rest is lost as waste heat.
Power Plants
Power plants are facilities designed to convert various forms of energy into electrical energy. The primary type referenced in the exercise is a coal-fired power plant, which utilizes coal as its fuel source. These plants operate based on the principle of converting chemical energy from coal into thermal energy, then into mechanical energy, and finally into electrical energy.
Types of power plants include:
  • Coal-fired plants: Utilize coal combustion to generate heat.
  • Nuclear plants: Use nuclear reactions to produce heat.
  • Hydroelectric plants: Convert the energy of flowing water into electricity.
  • Solar plants: Capture sunlight using photovoltaic cells or solar thermal methods.
Each type of power plant has a different efficiency and environmental impact, with coal plants being notorious for their heat emissions and carbon footprint.
Energy Conversion
Energy conversion is crucial in power generation. In thermodynamics, it involves changing energy from one form to another, such as from heat to mechanical.
In a coal-fired plant, the steps are:
  • Burning coal releases chemical energy, transforming into heat.
  • This thermal energy converts water into steam under high pressure.
  • The steam turns turbines, converting thermal energy to mechanical energy.
  • Finally, the mechanical energy spins generators, creating electric energy.
This series of energy conversions is subject to losses, mainly in the form of heat due to limited efficiency levels of each step
Therefore, improving any conversion process can enhance overall plant efficiency.
Heat Transfer
Heat transfer is a fundamental concept affecting power plant performance. It describes how heat moves from one area or material to another, a critical process in thermodynamic systems.
In power plants, heat transfer occurs mainly in three ways:
  • Conduction: Heat moves through a material without the material itself moving (e.g., through metal surfaces).
  • Convection: Heat is carried away by moving fluids or gases (e.g., steam in pipes).
  • Radiation: Heat is emitted as electromagnetic waves (normally negligible in boilers but significant elsewhere).
Effective management of heat transfer is vital for ensuring that as much energy as possible from the fuel is converted into electricity rather than lost to the environment.
This involves advanced insulation, efficient heat exchangers, and optimized boiler systems, all designed to minimize waste and maximize output.

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Most popular questions from this chapter

A cooling unit for chilling the water of an aquarium gives specifications of \(1 / 10 \mathrm{hp}\) and \(1270 \mathrm{Btu} / \mathrm{h}\). Assuming the unit produces its \(1 / 10\) hp at \(70.0 \%\) efficiency, calculate its performance coefficient.

A Carnot freezer that runs on electricity removes heat from the freezer compartment, which is at \(-10^{\circ} \mathrm{C},\) and expels it into the room at \(20^{\circ} \mathrm{C}\). You put an ice-cube tray containing \(375 \mathrm{~g}\) of water at \(18^{\circ} \mathrm{C}\) into the freezer. (a) What is the coefficient of performance of this freezer? (b) How much energy is needed to freeze this water? (c) How much electrical energy must be supplied to the freezer to freeze the water? (d) How much heat does the freezer expel into the room while freezing the ice?

A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{~J}\) of heat per minute from the room being cooled and in the same period deposits \(1.44 \times 10^{5} \mathrm{~J}\) of heat into the outside air. What is the power consumption of the unit in watts?

If the proposed plant is built and produces \(10 \mathrm{MW}\) but the rate at which waste heat is exhausted to the cold water is \(165 \mathrm{MW}\), what is the plant's actual efficiency? A. \(5.7 \%\) B. \(6.1 \%\) C. \(6.5 \%\) D. \(16.5 \%\)

A sophomore with nothing better to do adds heat to \(0.350 \mathrm{~kg}\) of ice at \(0.00^{\circ} \mathrm{C}\) until it is all melted. (a) What is the change in entropy of the water? (b) The source of the heat is a very massive body at a temperature of \(25.0^{\circ} \mathrm{C}\). What is the change in entropy of this body? (c) What is the total change in entropy of the water and the heat source?
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