Problem 48 - University Physics with Modern Physics

Question

A typical coal-fired power plant generates 1000 MW of usable power at an overall thermal efficiency of 40%. (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of 2.65 \(\times\) 10\(^7\) J/kg. How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river is at 18.0\(^\circ\)C before it reaches the power plant and 18.5\(^\circ\)C after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

Step-by-Step Solution

Answer

(a) 2500 MW heat input; (b) 8.16 million kg of coal/day; (c) 1500 MW heat ejected; (d) River flow rate: 716.6 m³/s; (e) Entropy increase: \(5.15 \times 10^6 \text{ J/K}\).

Step 1: Calculate the Heat Input Rate

To find the heat input rate, we use the formula for efficiency, \( \eta = \frac{P}{Q_{in}} \), where \( P = 1000 \text{ MW} \) is the power output and \( \eta = 0.4 \) is the efficiency. Solving for \( Q_{in} \):\[Q_{in} = \frac{P}{\eta} = \frac{1000 \times 10^6 \text{ W}}{0.4} = 2500 \times 10^6 \text{ W} = 2500 \text{ MW}\]Thus, the rate of heat input to the plant is 2500 MW.

Step 2: Calculate Coal Usage per Day

The energy obtained from coal is given by \( Q_{in} = m \cdot H_c \), where \( m \) is the mass of coal and \( H_c = 2.65 \times 10^7 \text{ J/kg} \). Therefore,\[m = \frac{Q_{in}}{H_c} = \frac{2500 \times 10^6 \text{ W} \times 86400 \text{ s}}{2.65 \times 10^7 \text{ J/kg}} = 8.16 \times 10^6 \text{ kg}\]So, the plant uses approximately \( 8.16 \text{ million kg} \) of coal per day.

Step 3: Calculate Heat Ejected Rate

The rate at which heat is ejected, \( Q_{out} \), can be found from \( Q_{in} = Q_{out} + P \). Rearranging gives:\[Q_{out} = Q_{in} - P = 2500 \times 10^6 \text{ W} - 1000 \times 10^6 \text{ W} = 1500 \times 10^6 \text{ W} = 1500 \text{ MW}\]Thus, the rate of heat ejected is 1500 MW.

Step 4: Calculate River Flow Rate

The rate of heat ejected into the river results in a temperature increase \( \Delta T = 0.5 \text{°C} \). The flow rate \( F \) is given by:\[Q_{out} = F \cdot c_p \cdot \Delta T\]where \( c_p = 4186 \text{ J/(kg°C)} \) is the specific heat capacity of water. Solving for \( F \):\[F = \frac{Q_{out}}{c_p \cdot \Delta T} = \frac{1500 \times 10^6}{4186 \cdot 0.5}\]\[F \approx 716.6 \times 10^3 \text{ kg/s} \approx 716.6 \text{ m}^3/\text{s}\]Thus, the river’s flow rate is approximately 716.6 m³/s.

Step 5: Calculate Increase in River's Entropy

The increase in entropy \( \Delta S \) of the river per second is\[\Delta S = \frac{Q_{out}}{T}\]where \( T = 291 \text{ K} \) (average temperature of the river in Kelvin). Substituting,\[\Delta S = \frac{1500 \times 10^6}{291} \approx 5.15 \times 10^6 \text{ J/K} \cdot \text{s}\]Therefore, the entropy increase per second is about \( 5.15 \times 10^6 \text{ J/K} \).

Key Concepts

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts fuel energy into usable electricity. In this context, the thermal efficiency of a power plant is given by the formula \( \eta = \frac{P}{Q_{in}} \), where \( P \) is the net power output and \( Q_{in} \) is the rate of heat input into the system. Here, the power plant operates at a thermal efficiency of 40%, meaning only 40% of the heat energy from burning coal is converted into electricity. This efficiency is crucial for determining the overall performance and cost-effectiveness of the power plant.
High thermal efficiency indicates a well-designed plant that uses less fuel input compared to the expired energy waste, reducing both cost and environmental impact.

Heat Transfer

Heat transfer in a power plant involves the transformation of energy from one form to another, primarily in three ways: conduction, convection, and radiation. In this scenario, heat is transferred from the combustion of coal to water, converting it into steam, which drives a turbine to generate electricity. The rate of heat transfer is vital for understanding how much heat is entering and leaving the system.

• Conduction: Heat flows through solid materials such as metal pipes.
• Convection: Heat is carried by fluid motion, like water or air flowing over surfaces.
• Radiation: Heat energy is emitted in the form of electromagnetic waves.

In this exercise, the concept of heat transfer helps us calculate how much energy is available for the power output and how much heat needs to be dissipated into the environment, affecting the efficiency and environmental safety.

Coal Combustion

Coal combustion is a chemical process where coal, primarily composed of carbon, burns in the presence of oxygen to release energy. Anthracite coal, used by the power plant in this scenario, is known for its high carbon content and energy density. It has a heat of combustion of \( 2.65 \times 10^7 \text{ J/kg} \), making it an efficient choice for generating large amounts of energy.
The process of combustion involves several stages:

• Preheating the coal to release volatile gases.
• Igniting these gases and the remaining solid carbon.
• Burning the material to produce heat, water vapor, CO\(_2\), and other by-products.

Efficient combustion minimizes the amount of unburnt fuel and emissions, directly affecting the plant's environmental footprint and energy efficiency.

Specific Heat Capacity

Specific heat capacity refers to the amount of heat required to change the temperature of a substance by one degree Celsius without altering its state. For the river receiving the power plant's waste heat, the specific heat capacity of water is approximately 4186 J/(kg°C). This property is essential for calculating how the temperature of the river changes as it absorbs heat.
The equation \( Q_{out} = F \cdot c_p \cdot \Delta T \) is used to calculate the river's flow rate by understanding how much heat the water can absorb and at what rate. The specific heat capacity allows us to predict the impact on the river’s temperature using the known increase of 0.5°C in this scenario. By understanding specific heat capacity, we can estimate how environmental systems cope with thermal discharges and maintain ecological stability.