Carnot Engine Efficiency Calculation How To Increase Efficiency By 50 Percent
The Carnot engine is a theoretical thermodynamic cycle that provides the maximum possible efficiency for converting heat into work. Understanding its principles is crucial for grasping the limitations and possibilities of real-world engines. This article delves into a problem involving a Carnot engine, exploring how changes in its source temperature affect its efficiency. We'll break down the concepts, calculations, and the underlying physics to provide a comprehensive understanding of the solution.
A Carnot engine operates with a sink temperature of 300 K and an initial efficiency of 40%. The question we aim to answer is: By how much should the temperature of the source be increased to raise the engine's efficiency by 50% of its original efficiency? The options provided are:
(A) 380 K
(B) 275 K
(C) 325 K
(D) 250 K
The efficiency (η) of a Carnot engine is determined by the temperatures of its heat source (T_H, the hot reservoir) and heat sink (T_C, the cold reservoir). The formula for Carnot efficiency is:
η = 1 - (T_C / T_H)
Where:
- η is the efficiency
- T_C is the absolute temperature of the cold reservoir (in Kelvin)
- T_H is the absolute temperature of the hot reservoir (in Kelvin)
This equation highlights a fundamental principle: the greater the temperature difference between the hot and cold reservoirs, the higher the engine's efficiency. A Carnot engine operating between two temperatures achieves the highest possible efficiency compared to any other engine operating between the same temperatures.
We're given the following initial conditions:
- Sink temperature (T_C) = 300 K
- Initial efficiency (η₁) = 40% or 0.4
We can use this information to calculate the initial source temperature (T_H₁). Plugging the values into the efficiency formula:
- 4 = 1 - (300 / T_H₁)
Rearranging the equation to solve for T_H₁:
- 4 = 300 / T_H₁
T_H₁ = 300 / 0.6
T_H₁ = 500 K
Therefore, the initial temperature of the source is 500 K.
The problem states that we need to increase the efficiency by 50% of the original efficiency. The original efficiency was 40%, so a 50% increase is:
50% of 40% = 0.50 * 0.4 = 0.2
The new target efficiency (η₂) is the initial efficiency plus this increase:
η₂ = η₁ + 0.2
η₂ = 0.4 + 0.2
η₂ = 0.6
So, the target efficiency is 60% or 0.6.
Now we need to find the new source temperature (T_H₂) required to achieve the target efficiency of 60%, while keeping the sink temperature constant at 300 K. Using the Carnot efficiency formula again:
- 6 = 1 - (300 / T_H₂)
Rearranging to solve for T_H₂:
- 4 = 300 / T_H₂
T_H₂ = 300 / 0.4
T_H₂ = 750 K
Thus, the new source temperature required to achieve a 60% efficiency is 750 K.
The problem asks for the increase in temperature, not the final temperature. To find the increase, we subtract the initial source temperature from the new source temperature:
Temperature increase = T_H₂ - T_H₁
Temperature increase = 750 K - 500 K
Temperature increase = 250 K
Therefore, the source temperature needs to be increased by 250 K to achieve the desired efficiency increase.
Based on our calculations, the correct answer is:
(D) 250 K
To summarize, the core concepts involved in solving this problem are:
- Carnot Engine Efficiency: The efficiency of a Carnot engine depends solely on the temperatures of its hot and cold reservoirs.
- Efficiency Formula: η = 1 - (T_C / T_H)
- Temperature Units: Absolute temperatures (Kelvin) must be used in the Carnot efficiency formula.
- Problem Decomposition: Breaking the problem into steps: finding the initial source temperature, calculating the target efficiency, finding the new source temperature, and finally, determining the temperature increase.
Understanding these concepts is essential for tackling similar problems involving thermodynamics and heat engines.
While the Carnot engine is an idealized model, it provides valuable insights into the design and performance of real-world engines. The Carnot efficiency sets an upper limit on the efficiency that any heat engine can achieve operating between two given temperatures. This understanding drives innovation in materials science, engine design, and other fields aimed at improving energy conversion efficiency.
When solving Carnot engine problems, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can improve accuracy:
- Using Celsius or Fahrenheit: Always convert temperatures to Kelvin before using them in the Carnot efficiency formula.
- Misinterpreting Efficiency Increase: Ensure you understand whether the problem asks for the new efficiency or the increase in efficiency.
- Incorrectly Rearranging the Formula: Practice rearranging the efficiency formula to solve for different variables (T_H, T_C, η).
- Not Reading the Question Carefully: Always double-check what the problem is specifically asking for (e.g., temperature increase vs. final temperature).
To deepen your understanding of thermodynamics and Carnot engines, consider exploring these topics further:
- Thermodynamic Cycles: Learn about other thermodynamic cycles, such as the Otto cycle (used in gasoline engines) and the Diesel cycle.
- Entropy: Understand the concept of entropy and its relationship to the second law of thermodynamics.
- Heat Engine Design: Research the design and operation of real-world heat engines, such as internal combustion engines and steam turbines.
By delving into these areas, you can develop a more comprehensive understanding of the principles governing energy conversion and the limitations and possibilities of various thermodynamic systems.
Solving the Carnot engine efficiency problem requires a clear understanding of the Carnot cycle, the efficiency formula, and careful attention to the problem statement. By systematically breaking down the problem into steps, we calculated the necessary temperature increase to achieve the target efficiency. This example demonstrates the importance of theoretical models like the Carnot engine in understanding the fundamental limits and possibilities of energy conversion. Mastering these concepts provides a solid foundation for further exploration in thermodynamics and related fields.
