Question

During January, at a location in Alaska winds at \(-30^{\circ} \mathrm{C}\) can be observed. Several meters below ground the temperature remains at \(13^{\circ} \mathrm{C}\), however. An inventor claims to have devised a power cycle exploiting this situation that has a thermal efficiency of \(10 \%\). Discuss this claim.

Short answer

The claim of 10% thermal efficiency is possible since it is less than the Carnot efficiency of 15.01%.

Step-by-step solution

- Understand Carnot Efficiency

The Carnot efficiency is given by the formula: \[ \text{Efficiency}_{\text{Carnot}} = 1 - \frac{T_L}{T_H} \]where \(T_L\) is the lower temperature of the cycle and \(T_H\) is the higher temperature of the cycle. These temperatures must be in absolute scale (Kelvin).

- Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin using the formula:\[T(K) = T(°C) + 273.15\]The temperatures in Kelvin are:\[T_L = -30 + 273.15 = 243.15 \text{ K} \]\[T_H = 13 + 273.15 = 286.15 \text{ K} \]

- Calculate Carnot Efficiency

Substitute the temperatures in Kelvin into the Carnot efficiency formula:\[ \text{Efficiency}_{\text{Carnot}} = 1 - \frac{243.15}{286.15} \approx 0.1501 \]This gives a Carnot efficiency of approximately 15.01%.

- Compare Claimed Efficiency with Carnot Efficiency

The inventor claims a thermal efficiency of 10%. Compare this with the calculated Carnot efficiency (15.01%). Since 10% is less than 15.01%, the claim is within the bounds of theoretical maximum efficiency for a heat engine operating between these temperatures.

Conclusion

The inventor's claim of a 10% thermal efficiency is possible and realistic, as it is below the Carnot efficiency limit of approximately 15.01%.

Key concepts

Thermal Efficiency

Thermal efficiency refers to the effectiveness of a heat engine in converting the heat energy it receives into useful work. In simple terms, it's a measure of how well an engine can transform thermal energy into mechanical energy. The formula for thermal efficiency (\text{η}) is given by: \[ \text{η} = \frac{W_{\text{out}}}{Q_{\text{in}}} \] where \[ W_{\text{out}} \] is the work output and \[ Q_{\text{in}} \] is the heat input.
For instance, if an engine has a thermal efficiency of 10%, it means only 10% of the heat energy is converted into work, while the remaining 90% is wasted. In our example, the inventor claims a thermal efficiency of 10%. By comparing this efficiency with the theoretical Carnot efficiency (which we calculated as approximately 15.01%), we can verify if the claim is realistic. The fact that 10% is less than 15.01% means the claim could be valid, highlighting how thermal efficiency helps in understanding and assessing the performance of heat engines.

Absolute Temperature

Absolute temperature is a fundamental concept in thermodynamics, where temperatures are measured in Kelvin (K) instead of degrees Celsius (°C) or Fahrenheit (°F).
This scale starts at absolute zero, the theoretical point where all molecular motion ceases. The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature, as shown in our example: \[ T(K) = T(°C) + 273.15 \] In the problem, the temperatures given were \[ -30^{\text{°C}} \] and \[ 13^{\text{°C}} \]. These convert to \[ 243.15 \text{ K} \] and \[ 286.15 \text{ K} \] respectively.
Understanding absolute temperature is crucial for calculating efficiencies, as the Carnot efficiency formula requires temperatures in Kelvin. This ensures all calculations are done on an absolute scale, providing accurate insights into the maximum possible efficiency of a heat engine.

Heat Engine Cycle

A heat engine operates on the principle of converting heat energy (usually from a high-temperature source) into mechanical work. This process typically involves a cycle of phases including heating, expansion, cooling, and compression.
The basic idea behind a heat engine cycle can be understood by examining the Carnot cycle, which is an idealized model representing the most efficient cycle possible between two temperature limits. The Carnot efficiency is calculated using the temperatures of the heat source and heat sink: \[ \text{Efficiency}_{\text{Carnot}} = 1 - \frac{T_L}{T_H} \] where \[ T_L \] is the lower temperature and \[ T_H \] is the higher temperature.
In our specific problem, the engine operates between \[ 243.15 \text{ K} \] and \[ 286.15 \text{ K} \], resulting in a Carnot efficiency of around 15.01%. Comparing this efficiency to the claimed 10% helps determine the cycle's feasibility. All practical heat engines have efficiencies less than the Carnot efficiency due to real-world losses like friction and heat dissipation. Understanding the heat engine cycle and its efficiencies thus offers valuable insights into the interplay between heat energy and mechanical work.