Finding an expression for the efficiency of a heat engine

In summary, the conversation discusses a problem related to finding the efficiency of a heat engine. The speaker is struggling to identify where their mistake is and asks for help. They mention that the answer key provided does not make sense and that a key factor in finding the correct answer is letting the cold reservoir have a temperature of 0 K. Another person confirms the original speaker's result and points out that the book's answer always results in a negative efficiency, which is not possible.
  • #1
guyvsdcsniper
264
37
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
gif.gif
.

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.

IMG_8465.JPG
 
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  • #2
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
  • #3
Thank you. I was breaking my head over this but letting Tc = 0k really does help illuminate the answer
 
  • #4
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).
 
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Likes hutchphd
  • #6
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.
 

1. What is a heat engine?

A heat engine is a device that converts heat energy into mechanical work. It typically consists of a heat source, a working fluid, and a mechanism to convert the energy into useful work.

2. How is efficiency defined for a heat engine?

The efficiency of a heat engine is defined as the ratio of the work output to the heat input. In other words, it measures how much of the input energy is converted into useful work.

3. What is the formula for calculating the efficiency of a heat engine?

The efficiency of a heat engine is calculated using the formula: efficiency = (work output / heat input) x 100%. This is also known as the Carnot efficiency formula.

4. What factors affect the efficiency of a heat engine?

The efficiency of a heat engine is affected by several factors, including the temperature of the heat source, the type of working fluid used, and the design and construction of the engine.

5. Can the efficiency of a heat engine ever be 100%?

No, according to the Second Law of Thermodynamics, it is impossible for a heat engine to have 100% efficiency. This is because some energy will always be lost as waste heat during the conversion process.

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