A Power plant rated at 800 MegaWatts generates steam at 585 K and discards heat to a river at 295 K. If thermal efficiency of plant is 70% of maximum value, how much heat is discarded to river at rated power?
Please help with steps on how to solve this one.
To find out how much heat is discarded to the river at rated power, we need to calculate the total heat input to the power plant and then deduct the useful work output.
Step 1: Calculate the total heat input to the power plant.
We know that the thermal efficiency of the power plant is 70% of the maximum value. Therefore, the actual efficiency is 0.70 times the maximum efficiency.
Efficiency = Useful Work Output / Total Heat Input
We can rearrange this equation to find the Total Heat Input:
Total Heat Input = Useful Work Output / Efficiency
Since the question doesn't mention any useful work output, we can assume that all the work done by the plant is useful and neglect any losses due to friction, for example.
Hence, the Total Heat Input = Rated Power = 800 MW.
Step 2: Find the useful work output.
To find the useful work output, we can use the formula:
Useful Work Output = Total Heat Input - Heat Discarded
Step 3: Find the heat discarded to the river at the rated power.
To calculate the heat discarded to the river, we can use the formula for heat transfer:
Heat Transfer = Mass Flow Rate × Specific Heat Capacity × Temperature Difference
Given that the temperature of the steam is 585 K and the temperature of the river is 295 K, we'll use these values to calculate the temperature difference.
Step 4: Calculate the temperature difference.
Temperature Difference = Steam Temperature - River Temperature
Temperature Difference = 585 K - 295 K = 290 K
Step 5: Calculate the heat discarded to the river.
Heat Discarded = Heat Transfer = Rated Power = 800 MW
So, at rated power, 800 MW of heat is discarded to the river.
To solve this problem, we need to use the concept of thermal efficiency and the principles of thermodynamics. Here are the step-by-step calculations:
Step 1: Understand the Concept of Thermal Efficiency
Thermal efficiency is defined as the ratio of the useful work done or energy output to the energy input. It is given by the formula:
Thermal efficiency = (Useful work done or energy output / Energy input) * 100%
Step 2: Calculate the Maximum Theoretical Efficiency
The maximum theoretical efficiency of a heat engine operating between two temperature reservoirs can be calculated using the Carnot efficiency formula:
Max Efficiency (Carnot efficiency) = 1 - (Thot / Tcold)
where:
Thot = Hot reservoir temperature (in Kelvin)
Tcold = Cold reservoir temperature (in Kelvin)
Step 3: Calculate the Maximum Theoretical Efficiency at 585 K and 295 K
Using the given temperatures, we can calculate the maximum theoretical efficiency (Carnot efficiency) of the power plant:
Max Efficiency = 1 - (585 K / 295 K)
Step 4: Calculate the Thermal Efficiency of the Power Plant
Since the problem states that the thermal efficiency of the power plant is 70% of the maximum theoretical value, we need to calculate this value:
Thermal Efficiency = 70% * Max Efficiency
Step 5: Calculate the Energy Input to the Power Plant
The energy input to the power plant can be calculated using the formula:
Energy input = Energy output / Thermal Efficiency
Step 6: Calculate the Heat Discarded to the River
The heat discarded to the river can be found by subtracting the energy output from the energy input:
Heat Discarded to River = Energy input - Energy output
Now let's calculate the values:
Step 3 Calculation:
Max Efficiency = 1 - (585 K / 295 K)
Max Efficiency = 1 - 1.983
Step 4 Calculation:
Thermal Efficiency = 0.7 * Max Efficiency
Step 5 Calculation:
Energy Input = Energy Output / Thermal Efficiency
Step 6 Calculation:
Heat Discarded to River = Energy Input - Energy Output
Given that the power plant is rated at 800 MegaWatts, it means that the energy output is 800 MW.
Hope this helps!