# Carnot Cycle and Line Integrals

• longbusy
In summary, the Carnot cycle and line integrals can be related in terms of finding the work done by a vector field on an object traveling along a certain path. The Carnot cycle also sets a limit on the efficiency of an engine cycle. To compare them more specifically, you could consider the effect of heat on the system. However, it should be noted that while the Carnot cycle is often used as an ideal representation for engine cycles, real world engines typically use different cycles such as the Otto or Diesel cycles. Additionally, the Carnot cycle is only applicable to reversible transformations, and its limitations are expressed through the Clausius Theorem, which is based on line integrals. Further research should be done using a reputable thermodynamics textbook

#### longbusy

I have to corellate the Carnot cycle with line integrals. This makes sense to me as line integrals can be used to find the work done by a vector field on an object traveling along a certain path. The Carnot cycle places a limit on the efficiency of an engine cycle.

My question, how could I specifically compare line integrals to the Carnot cycle. Maybe the effect that heat has on the system? I'll attach what I have written so far. Thanks!

(Edit: I attached the .doc file but I do not see it on my post.)

#### Attachments

• carnot cycle.doc
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You could try using Green's theorem to relate the Line Integrals around the cycle to the area inside the region, however since when using Green's theorem you usually go around the region counterclockwise it would be in the wrong direction of your cylce.

Originally posted by longbusy
I have to corellate the Carnot cycle with line integrals. This makes sense to me as line integrals can be used to find the work done by a vector field on an object traveling along a certain path. The Carnot cycle places a limit on the efficiency of an engine cycle.

My question, how could I specifically compare line integrals to the Carnot cycle. Maybe the effect that heat has on the system? I'll attach what I have written so far. Thanks!

You might want to start having a look at http://scienceworld.wolfram.com/physics/topics/ThermodynamicCycles.html. [Broken]
The first remark on your doc is that you are assuming car engines run on a Carnot Cycle which is not true since gasoline cars use Otto cycle and diesel cars use Diesel cycle.

Besides that you have to consider that Carnot Cycle is an ideal representation valid for quasi-static (reversible) transformations. This ideal representation set a limit to real world engines through the so called Clausius Theorem that is based on line integrals.

I will write you more tomorrow... search the internet for a decent thermodynamic textbook.

DArio

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## 1. What is the Carnot Cycle?

The Carnot Cycle is a theoretical thermodynamic cycle that describes the most efficient way to convert heat into work or vice versa. It consists of four reversible processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

## 2. How does the Carnot Cycle relate to heat engines?

The Carnot Cycle is the basis for the operation of heat engines, as it describes the maximum efficiency that can be achieved in converting heat into work. Real-life heat engines are not able to reach the theoretical maximum efficiency due to various factors such as friction and energy losses.

## 3. What is the significance of the Carnot Cycle in thermodynamics?

The Carnot Cycle is significant because it provides a theoretical framework for understanding the limitations and efficiency of heat engines. It also serves as a benchmark for comparing the performance of real-life heat engines.

## 4. What are line integrals and how are they related to the Carnot Cycle?

Line integrals are mathematical tools used to calculate the work done by a vector field along a path. In the context of the Carnot Cycle, line integrals can be used to calculate the work done by the system during each step of the cycle, as well as the total work done over the entire cycle.

## 5. How does the Carnot Cycle demonstrate the second law of thermodynamics?

The second law of thermodynamics states that heat always flows from a hot object to a cold object, and that the total entropy of an isolated system will never decrease. The Carnot Cycle demonstrates this law through its reversible processes, which maintain a constant temperature gradient and result in zero net change in entropy over a full cycle.