Finding an expression for the efficiency of a heat engine

AI Thread Summary
The discussion focuses on finding the correct expression for the efficiency of a heat engine, with participants noting discrepancies between their calculations and the provided answer key. A key point raised is the confusion regarding the variables Tc/th and tc/tm, which appear to be flipped. Participants emphasize that if the cold reservoir is set to 0 K, the expected efficiency should be 100%, a result not reflected in the answer key. Additionally, it is highlighted that the book's answer yields negative efficiency values, which are deemed incorrect. The conversation underscores the need for clarity in algebraic manipulation to resolve these issues.
guyvsdcsniper
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Homework Statement
A reversible heat engine extracts
heat QH > 0 from a reservoir at temperature TH and heat QM = aQH > 0
from a reservoir at temperature TM ≤ TH while rejecting waste heat QC > 0
to a reservoir at temperature TC ≤ TM.

Derive an expression for the effi ciency of this three- reservoir heat
engine in terms of a and the three temperatures TH, TM, and TC ,
where the effi ciency is the total work produced divided by the total
heat extracted from the two hotter reservoirs.
Relevant Equations
(1 − TH /TC )/(1 + a) + (1 − TM /TC )a/(1 + a)
My book states the answer to this problem is
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.

I have gotten very close to the answer. My problem is my Tc/th and tc/tm are flipped compared to the solution.

I feel like I am missing something in my algebra but can't see where I am going wrong. Could I get some help identifying where my mistake is.

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The answer key wherever it's coming from doesn't make sense. Remember if the cold reservoir has 0 K, you are supposed to get 100% efficiency which is not happening with the answer key you cited.
 
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Likes vela, Steve4Physics and guyvsdcsniper
Thank you. I was breaking my head over this but letting Tc = 0k really does help illuminate the answer
 
Another handy check is to let ##a=0##. This gives a simple heat engine working between ##T_h## and ##T_c##. The efficiency should then be ##1 - \frac {T_c}{T_h}## (standard formula).
 
guv said:
The answer key wherever it's coming from doesn't make sense. Remember if the cold reservoir has 0 K, you are supposed to get 100% efficiency which is not happening with the answer key you cited.
You might also notice that the book's answer always results in a negative number for the efficiency. It can't possibly be correct.
 
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