Question

A patent application describes a device that at steady state receives a heat transfer at the rate \(1 \mathrm{~kW}\) at a temperature of \(167^{\circ} \mathrm{C}\) and generates electricity. There are no other energy transfers. Does the claimed performance violate any principles of thermodynamics? Explain.

Short answer

Yes, the claimed performance violates the second law of thermodynamics as it assumes 100% conversion of heat to work, which is impossible.

Step-by-step solution

- Identify the Working Temperatures

Convert given temperature to Kelvin. Temperature in Celsius is given as 167°C. Use the formula: \[ T(K) = T(°C) + 273.15 \] So, the temperature in Kelvin is \[ 167 + 273.15 = 440.15 \text{ K} \]

- Determine the Heat Transfer Rate

Given the heat transfer rate is 1 kW (which is equivalent to 1000 W).

- Understand the Claimed Performance

The device claims to generate electricity by receiving heat at the given rate.

- Apply the Second Law of Thermodynamics

According to the second law of thermodynamics, no device can convert heat completely into work without any other effect. An ideal heat engine's efficiency is given by: \[ \text{Efficiency} = 1 - \frac{T_C}{T_H} \] where \( T_H \) is the high-temperature reservoir (in this case, 440.15 K) and \( T_C \) is the low-temperature reservoir (usually, the ambient temperature). Assume ambient temperature \( T_C \) is 300 K.

- Calculate the Maximum Efficiency

Using the temperatures: \[ \text{Efficiency} = 1 - \frac{300}{440.15} = 1 - 0.6815 = 0.3185 \text{ or } 31.85\text{\text%} \]

- Evaluate the Performance

The maximum theoretical efficiency is 31.85%, meaning the generated electricity can be at most 31.85% of the heat received. For a 1 kW heat transfer: \[ \text{Max Electricity Generation} = 1 \text{ kW} \times 0.3185 = 0.3185 \text{ kW} \text{ or } 318.5 \text{ W} \]

- Conclusion

If the patent claims converting 1 kW of heat directly into 1 kW of electricity, it violates the second law of thermodynamics because it exceeds the maximum possible efficiency.

Key concepts

heat engine efficiency

Heat engine efficiency is a key concept in thermodynamics. It measures how well a heat engine converts heat into useful work. The efficiency of a heat engine is defined as the ratio of work output to heat input. The formula for the efficiency of an ideal heat engine, also known as the Carnot efficiency, is given by:
\[ \text{Efficiency} = 1 - \frac{T_C}{T_H} \]
Here, \(T_H\) represents the temperature of the high-temperature reservoir, and \(T_C\) denotes the temperature of the low-temperature reservoir.

According to the second law of thermodynamics, no heat engine can have an efficiency of 100%, meaning that some heat must always be rejected to the surroundings. In the example given, the temperatures are 440.15 K (from 167°C) and 300 K (ambient temperature).

Using the formula, we find that the maximum efficiency is approximately 31.85%. Thus, the device in the problem can theoretically convert only up to 31.85% of the input heat into electricity. This translates to a maximum electricity generation of around 318.5 W from an input of 1 kW.

thermodynamic principles

Thermodynamic principles are the laws that govern the transfer of energy and the direction of these processes. The Second Law of Thermodynamics states that heat cannot spontaneously flow from a colder body to a hotter body, and also that it is impossible to convert all the heat from a heat source into work.

For any heat engine, this law implies that its efficiency will always be less than 100%. This principle is crucial when assessing the performance of heat engines and other devices that convert energy.

In the provided exercise, the second law directly addresses the claimed performance of the device. If a patent claims that a device can convert 1 kW of heat into 1 kW of electricity without any heat rejection, it violates this law. The actual efficiency calculated (31.85%) showcases that the claimed performance is not feasible and emphasizes the importance of the second law in evaluating such systems.

heat transfer

Heat transfer is the process of thermal energy moving from one body or system to another. It occurs via conduction, convection, or radiation. In the context of thermodynamics, heat transfer plays a significant role in the performance of heat engines.

In heat engines, heat is usually transferred from a high-temperature source to a working substance, which then performs work by moving a piston or turning a turbine. The remaining heat is transferred to a low-temperature sink. The efficiency of a heat engine is determined by how well it can utilize the heat transfer processes.

In the described exercise, the device receives heat at a rate of 1 kW. This heat input is essential for generating electricity. However, laws of thermodynamics stipulate that not all heat can be used for generating work; some of it must be rejected to the surroundings. This highlights the inherent limitations due to heat transfer inefficiencies, further underscoring the impossibility of the claims made by the patent.

• High-temperature reservoir: Where heat is absorbed.
• Working substance: Converts heat into work.
• Low-temperature sink: Where excess heat is rejected.