To determine the efficiency of the steam engine, the amount of work it produces, the heat it receives, and the heat wasted must be analyzed. Let's break this down step by step.
Calculating Efficiency
The efficiency of an engine can be calculated using the formula:
Efficiency (η) = (Work Output / Heat Input) × 100%
In this case, the work output is given as:
Work Output = 7.5 × 108 J/min
The heat input from the boiler is:
Heat Input = 3.6 × 109 J/min
Now, substituting these values into the efficiency formula:
Efficiency (η) = (7.5 × 108 J/min) / (3.6 × 109 J/min) × 100%
Calculating this gives:
η = (7.5 / 36) × 100% = 20.83%
Heat Wasted Calculation
To find out how much heat is wasted per minute, we can use the following relationship:
Heat Wasted = Heat Input - Work Output
Substituting the known values:
Heat Wasted = 3.6 × 109 J/min - 7.5 × 108 J/min
Calculating this gives:
Heat Wasted = 3.6 × 109 J/min - 0.75 × 109 J/min = 2.85 × 109 J/min
Finding the Temperature Ratio
To find the ratio of the temperatures of the sink (Tsink) to the source (Tsource
Efficiency (η) = 1 - (Tsink / Tsource)
Rearranging this gives:
Tsink / Tsource = 1 - η
Substituting the efficiency we calculated:
Tsink / Tsource = 1 - 0.2083 = 0.7917
This means that the ratio of the temperature of the sink to the source is approximately:
0.7917
Summary of Findings
- Efficiency of the engine: 20.83%
- Heat wasted per minute: 2.85 × 109 J/min
- Temperature ratio (Tsink / Tsource): 0.7917
This analysis shows how we can derive important performance metrics for a steam engine, illustrating the principles of thermodynamics in action.
