Question

We can think of wind in the atmosphere as a giant heat engine \(^{67}\) operating between the \(288 \mathrm{~K}\) surface and the top of the troposphere \(^{68}\) at \(230 \mathrm{~K}\). What is the maximum efficiency this heat engine could achieve in converting solar heating into airflow?

Short answer

Answer: The maximum efficiency of the heat engine is approximately 20.14%.

Step-by-step solution

Identify the given information

We are given the two temperatures between which the heat engine operates: - \(T_H = 288 \mathrm{~K}\) (higher temperature - surface) - \(T_C = 230 \mathrm{~K}\) (lower temperature - top of the troposphere)

Write down the formula for the Carnot efficiency

The formula for the efficiency of a heat engine based on the Carnot cycle is: \(\eta = 1 - \frac{T_C}{T_H}\)

Substitute the given temperatures into the formula

We substitute the given temperatures into the formula for the Carnot efficiency: \(\eta = 1 - \frac{230 \mathrm{~K}}{288 \mathrm{~K}}\)

Calculate the efficiency

We perform the calculation to find the efficiency: \(\eta = 1 - \frac{230}{288} = 1 - 0.7986 \approx 0.2014\)

Express the efficiency as a percentage

To express the efficiency as a percentage, we multiply the result by 100: \(\eta = 0.2014 \times 100\% = 20.14\%\) The maximum efficiency this heat engine could achieve in converting solar heating into airflow is approximately 20.14%.

Key concepts

Heat Engine

A heat engine is a fascinating concept in thermodynamics. It's a system that converts heat energy into mechanical work, using a cycle of processes that typically involve heat transfer from a high-temperature source to a lower-temperature sink.
This cycle involves different stages of heating, expansion, cooling, and compression. It mimics processes found in natural and artificial settings, enabling us to harness energy for practical applications.

• The primary function of a heat engine is to produce work by using thermal energy.
• It operates between two reservoirs: a high-temperature source \(T_H\) and a low-temperature sink \(T_C\).
• The efficiency of a heat engine is determined by its ability to convert as much heat as possible into work.

Understanding heat engines helps us appreciate the natural and engineered systems around us. For instance, power plants and car engines rely on similar principles, but so does the wind flow in our atmosphere.

Carnot Efficiency

The Carnot efficiency represents the theoretical maximum efficiency a heat engine can achieve operating between two temperature reservoirs. Named after Sadi Carnot, it is derived from the second law of thermodynamics.

• The Carnot efficiency formula is: \( \eta = 1 - \frac{T_C}{T_H} \).
• \(T_H\) is the absolute temperature of the heat source, and \(T_C\) is the absolute temperature of the heat sink.

This formula sets an upper boundary on how efficiently a heat engine can possibly operate. It shows us that the efficiency increases as the temperature difference \((T_H - T_C)\) increases.
Applying this to atmospheric science, Carnot efficiency helps in predicting how efficiently solar energy can convert into wind energy. This is crucial for understanding weather patterns and atmospheric circulation.

Atmospheric Science

Atmospheric science often involves studying how energy from the sun influences the Earth's weather and climate. The atmosphere acts similarly to a massive heat engine, where solar energy is absorbed at the Earth's surface and then redistributed.

• The differential heating of Earth's surface affects the atmospheric conditions, creating winds and driving weather patterns.
• The Carnot efficiency, in this context, helps determine the potential effectiveness of solar energy conversion into atmospheric motions.

Understanding this efficiency is significant because it allows meteorologists to predict weather systems and climate patterns better. It's a crucial part of climate modeling, helping scientists gauge how efficiently energy transitions into winds, storms, and various circulation patterns.
By simulating maximum efficiencies, researchers can advance our understanding of climatological phenomena and improve weather forecasting accuracy.