A coal-fired electrical generating plant has an efficiency of 38%. The temperature of the steam leaving the boiler is 550 °C. What percentage of the maximum possible efficiency does this plant obtain? (Assume the temperature of the environment is 20 °C)
To determine the percentage of the maximum possible efficiency, we need to compare the actual efficiency of the plant with the maximum theoretical efficiency.
The maximum theoretical efficiency can be calculated using the formula:
Maximum theoretical efficiency = 1 - (T2 / T1)
Where:
T1 = Temperature of the steam leaving the boiler
T2 = Temperature of the environment
In this case:
T1 = 550 °C
T2 = 20 °C
Plugging in the values:
Maximum theoretical efficiency = 1 - (20 / 550)
Maximum theoretical efficiency ≈ 0.9636 (rounded to four significant figures)
Now, let's calculate the percentage of the maximum possible efficiency the plant obtains:
Percentage of maximum possible efficiency = (Actual efficiency / Maximum theoretical efficiency) * 100
Given that the actual efficiency of the plant is 38%:
Percentage of maximum possible efficiency = (0.38 / 0.9636) * 100
Percentage of maximum possible efficiency ≈ 39.44% (rounded to two decimal places)
Therefore, the coal-fired electrical generating plant obtains approximately 39.44% of the maximum possible efficiency.
To determine the percentage of the maximum possible efficiency that the coal-fired electrical generating plant obtains, we need to compare its efficiency with the maximum Carnot efficiency for the given temperature conditions.
The maximum Carnot efficiency of a heat engine is given by the formula:
Efficiency_max = (T_hot - T_cold) / T_hot
Where:
- T_hot is the temperature of the heat source (in this case, the temperature of the steam leaving the boiler)
- T_cold is the temperature of the heat sink (in this case, the temperature of the environment)
Given:
- T_hot = 550 °C
- T_cold = 20 °C
Plugging the values into the formula, we can calculate the maximum Carnot efficiency:
Efficiency_max = (550 - 20) / 550
Efficiency_max = 530 / 550
Efficiency_max = 0.9645
Therefore, the maximum Carnot efficiency for the given temperature conditions is approximately 0.9645 or 96.45%.
Now, to determine the percentage of the maximum possible efficiency that the plant obtains, we simply divide its efficiency by the maximum Carnot efficiency and multiply by 100:
Percentage_of_max_efficiency = (Plant_efficiency / Efficiency_max) * 100
Given:
- Plant_efficiency = 38%
Plugging the values into the formula, we can calculate:
Percentage_of_max_efficiency = (0.38 / 0.9645) * 100
Percentage_of_max_efficiency = 39.41%
Therefore, the coal-fired electrical generating plant obtains approximately 39.41% of the maximum possible efficiency for the given temperature conditions.