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If I consider a heat engine H and a refrigerator R such that:

where Q1<Q'1 and Q2<Q'2. Now if we were to connect the engine and the refrigerator in the following way:   We notice that Q1=W+Q2 and Q'1=W+Q'2 by the conservation of energy. From some manipulations of these equations we arrive at: Q'1-Q1=Q'2-Q2. Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that:

Where I have defined Q=Q'1-Q1=Q'2-Q2. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(When I asked the above question in other places, the response was that just like perpetual machines, such a configuration, although might seem to work on paper, cannot exist in real life. Is this what the answer should be?)

If I consider a heat engine $\mathrm{H}$ and a refrigerator $\mathrm{R}$ such that:

where $Q_1<Q'_1$ and $Q_2<Q'_2$.

Now if we were to connect the engine and the refrigerator in the following way  : 

We notice that :

(a) $Q_1=W+Q_2$

(b) $Q'_1=W+Q'_2$

(by the conservation of energy)

From some manipulations of these equations, we arrive at  :

$$Q'_1-Q_1=Q'_2-Q_2$$

Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that  :

Where I have defined $Q=Q'_1-Q_1=Q'_2-Q_2$. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(When I asked the above question in other places, the response was that just like perpetual machines, such a configuration, although might seem to work on paper, cannot exist in real life. Is this what the answer should be?)

If I consider a heat engine H and a refrigerator R such that:

where Q1<Q'1 and Q2<Q'2. Now if we were to connect the engine and the refrigerator in the following way: We notice that Q1=W+Q2 and Q'1=W+Q'2 by the conservation of energy. From some manipulations of these equations we arrive at: Q'1-Q1=Q'2-Q2. Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that:

Where I have defined Q=Q'1-Q1=Q'2-Q2. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(I expect to have made a very trivial error somewhere in my argument, but if you could point it out it would be of great help.)

If I consider a heat engine H and a refrigerator R such that:

where Q1<Q'1 and Q2<Q'2. Now if we were to connect the engine and the refrigerator in the following way: We notice that Q1=W+Q2 and Q'1=W+Q'2 by the conservation of energy. From some manipulations of these equations we arrive at: Q'1-Q1=Q'2-Q2. Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that:

Where I have defined Q=Q'1-Q1=Q'2-Q2. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(When I asked the above question in other places, the response was that just like perpetual machines, such a configuration, although might seem to work on paper, cannot exist in real life. Is this what the answer should be?)

If I consider a heat engine H and a refrigerator R such that:

where Q1<Q'1 and Q2<Q'2. Now if we were to connect the engine and the refrigerator in the following way: We notice that Q1=W+Q2 and Q'1+W+Q'2 by the conservation of energy. From some manipulations of these equations we arrive at: Q'1-Q1=Q'2-Q2. Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that:

Where I have defined Q=Q'1-Q1=Q'2-Q2. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(I expect to have made a very trivial error somewhere in my argument, but if you could point it out it would be of great help.)

If I consider a heat engine H and a refrigerator R such that:

where Q1<Q'1 and Q2<Q'2. Now if we were to connect the engine and the refrigerator in the following way: We notice that Q1=W+Q2 and Q'1=W+Q'2 by the conservation of energy. From some manipulations of these equations we arrive at: Q'1-Q1=Q'2-Q2. Now if we were to draw a box around the refrigerator and the engine and in turn call it the new refrigerator, we see that:

Where I have defined Q=Q'1-Q1=Q'2-Q2. The resulting refrigerator that we get has the ability to transfer heat from the hot body to the cold body without the help of any external work, which is a violation of the second law of thermodynamics.

(I expect to have made a very trivial error somewhere in my argument, but if you could point it out it would be of great help.)