# Engine working between multiple temperature baths worse than Carnot

• Sat D
In summary, the conversation discusses the construction of a composite engine, CmE, consisting of a new engine (NE) and a reverse Carnot engine (CE). NE extracts heat from a subset of baths and performs an amount of work, while CE operates between different temperatures and takes in heat from one end and releases it at the other. The goal is to show that the efficiency of CmE is less than that of NE alone, based on the reasoning that no other engine operating between two temperatures can be more efficient than a Carnot engine.
Sat D
Homework Statement
Imagine a standard heat engine running at multiple temperature baths of temperature ##T_{max}=T_1>T_2>T_3>...>T_n=T_{min}##. It extracts heat from a certain subset of these baths and rejects heat to the others. I want to prove that a Carnot engine running between ##T_1## and ##T_n## is more efficient than this engine.
Relevant Equations
##\eta_{Carnot} = 1-\frac{T_c}{T_h}##
Conservation of energy
Let the new engine, NE, extract heat from a certain subset of these baths, and let heat obtained from the ##i^{\rm th}## bath be denoted by ##Q_i##, and let the heat rejected to the ##j^{\rm th}## be denoted by ##Q_j##. Let the engine perform an amount of work ##W##.

Now right beside this engine, I construct a reverse Carnot engine, CE, operating between ##T_1## and ##T_n## such that it takes the work NE performed. CE takes in heat ##Q_c '## from ##T_n## and releases heat ##Q_h '## to ##T_1##.

I now wrap NE and CE into a composite engine, let's call it CmE. PE operates between temperatures ##T_i## and ##T_j##, where it takes heat ##\sum _i Q_i - Q_h'## and releases heat ##\sum_j Q_j - Q_c'##.

Since $$\sum _i Q_i - Q_h ' > 0 \implies \sum _i Q_i > Q_h '$$

Dividing both sides by ##W## and inverting them,
$$\frac{W}{\sum _i Q_i} < \frac{W}{Q_h'} \implies \eta _{_{NE}} < \eta _{_{CE}}$$
However, is this a valid proof? I base this off a proof I saw about how no other engine operating between two temperatures ##T_h## and ##T_c## can be more efficient than a Carnot engine, and the proof followed the same reasoning, just without the summation symbols.

Any advice would be appreciated.

Seems to me a "standard" Carnot engine runs "forward"...my interpretation. You know the efficiency of a single engine...what is the efficiency of two "coupled" engines (this is simple)? Show that is less than a single engine.
You don't need to rederive Carnot...but should be able to!

## 1. How does an engine working between multiple temperature baths compare to a Carnot engine?

An engine working between multiple temperature baths is less efficient than a Carnot engine. This is because a Carnot engine operates at the highest possible efficiency for a given temperature difference, while an engine working between multiple temperature baths has to work with varying temperature differences, resulting in lower efficiency.

## 2. Why is an engine working between multiple temperature baths less efficient?

An engine working between multiple temperature baths is less efficient because it has to work with varying temperature differences, which leads to a decrease in efficiency compared to a Carnot engine that operates at the highest possible efficiency for a given temperature difference.

## 3. Can an engine working between multiple temperature baths ever be as efficient as a Carnot engine?

No, an engine working between multiple temperature baths can never be as efficient as a Carnot engine. This is because a Carnot engine operates at the theoretical maximum efficiency for a given temperature difference, while an engine working between multiple temperature baths has to deal with practical limitations and varying temperature differences.

## 4. What are some practical applications where an engine working between multiple temperature baths is used?

An engine working between multiple temperature baths can be found in various practical applications, such as refrigerators, heat pumps, and power plants. These systems use multiple temperature baths to transfer heat and generate work, but they are less efficient than a Carnot engine.

## 5. Is there any way to improve the efficiency of an engine working between multiple temperature baths?

There are various ways to improve the efficiency of an engine working between multiple temperature baths, such as using more efficient materials, optimizing the design, and reducing energy losses. However, it will never reach the theoretical maximum efficiency of a Carnot engine due to practical limitations.

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