Finding an expression for the efficiency of a heat engine

[guyvsdcsniper]

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

.

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.

 

[guv]

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.

 

[guyvsdcsniper]

Thank you. I was breaking my head over this but letting Tc = 0k really does help illuminate the answer

 

[Steve4Physics]

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).

 

[Chestermiller]

I confirm your result.

 

[vela]

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.