Thermodynamics problem, Heat Engine with two heat sinks

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Discussion Overview

The discussion revolves around a thermodynamics problem involving a heat engine with one heat source and two heat sinks. Participants explore how to set up equations to determine the maximum thermal efficiency given the temperatures of the heat source and sinks, as well as the condition that the heat rejected to both sinks is equal.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant struggles to set up the equations for the heat engine problem, noting the temperatures of the heat source and sinks and the condition of equal heat rejection.
  • Another participant suggests calculating the maximum thermal efficiency for two separate systems and multiplying the results, questioning the relevance of the equal heat rejection condition.
  • A participant challenges the assumption that the heat rejected can simply be summed as QH = 2QC, asking about the role of work (W) in the equations.
  • Some participants propose using the first and second laws of thermodynamics to approach the problem, suggesting that the Carnot cycle may not be directly applicable due to the presence of three temperatures.
  • One participant expresses gratitude for the assistance received and mentions successfully formulating an equation for maximum Carnot efficiency by considering all three temperatures.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach to the problem. There are competing views on how to handle the equal heat rejection condition and whether to apply the Carnot efficiency directly.

Contextual Notes

Some participants note the absence of a specified work term (W) in the problem, which affects their calculations. There is also uncertainty regarding the application of the Carnot cycle due to the involvement of multiple temperatures.

TeddyLu
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I'm having a hard time setting up an equation for a heat engine problem with one heat source and two heat sinks given only the temperature of the heat source and temperature of the two heat sinks as:

TH = 1000 K
TC1 = 200 K
TC2 = 300 K

It is given that the two heat rejected are of equal value.

determine the maximum thermal efficiency.

Homework Equations


thermal efficiency (carnot) = 1 - TC/TH = 1 - QC/QH

The Attempt at a Solution


Since there was two heat sinks with an equal value of heat rejected at each:
QC1 = QC2 = QC

therefore, QH = QC1 + QC2 will turn into
QH = 2QC

I took the entropy balance equation to solve for QC:

0 = QH/1000 - QC/200 - QC/300

but I don't have a constant on the other side to figure out for QC to plug back into find QH and then solve for thermal efficiency.

any help please?
 
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Consider 2 systems First one with heat source at 1000 K and heat sink at 300 K. Find out the maximum thermal efficiency. For the second one, consider the source at 300 K and sink at 200 K. Calculate the maximum efficiency. Multiply the two to get the overall maximum efficiency.

I kept wondering why the equal rejected fact is mentioned. Couldn't figure it out. I'm not too sure this method is correct. Verify it from someone.
 
TeddyLu said:
therefore, QH = QC1 + QC2 will turn into
QH = 2QC
How did you get this? What happened to W?

You can use the 1st and 2nd laws to solve this problem. The answer is a function of all three temperatures, as you might guess.
I would start from 1st principles rather than attempt to squeeze the Carnot law into the problem. A Carnot cycle runs between 2 temperatures only, by definition.
 
@siddharth23
Thank you for your reply! I found out that rude man's suggestion below helped with finding the answer to my question.

@rude man
Thank you also for your reply. To answer your question, I used the entropy balance equation to comes to that conclusion. There was no W stated so I excluded it from the problem.
I followed your instructions and was able to formulate an equation to find out the maximum carnot efficiency. The use of all three temperatures and made sure to take half of the heat rejected (QC1 & QC2) and solved for QC/QH to plug back into my equation for the carnot efficiency.

Appreciate all the help for this problem!
 
TeddyLu said:
@siddharth23
Thank you for your reply! I found out that rude man's suggestion below helped with finding the answer to my question.

@rude man
Thank you also for your reply. To answer your question, I used the entropy balance equation to comes to that conclusion. There was no W stated so I excluded it from the problem.
I followed your instructions and was able to formulate an equation to find out the maximum carnot efficiency. The use of all three temperatures and made sure to take half of the heat rejected (QC1 & QC2) and solved for QC/QH to plug back into my equation for the carnot efficiency.

Appreciate all the help for this problem!
Good. So what was your answer?
 

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