Thermodynamics problem, Heat Engine with two heat sinks

In summary: It should be a single number, not a formula.In summary, the maximum thermal efficiency for the heat engine problem with one heat source and two heat sinks, given temperatures of the heat source and heat sinks, is determined by multiplying the efficiencies of two systems where the heat source is at the highest temperature and the heat sink is at the lowest temperature. The overall maximum efficiency is then calculated by taking half of the heat rejected from each heat sink and solving for QC/QH to plug into the Carnot efficiency equation. The result is a single number.
  • #1
TeddyLu
8
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
@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!
 
  • #5
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?
 

1. What is a heat engine?

A heat engine is a device that converts heat energy into mechanical work. It operates by taking in heat from a high temperature source, converting some of this energy into work, and then releasing the remaining heat to a lower temperature sink.

2. How does a heat engine work?

A heat engine works by utilizing the principles of thermodynamics. It operates in a cyclical process, where heat is absorbed from a high temperature source, work is performed, and the remaining heat is released to a lower temperature sink. This process is repeated to continuously produce work.

3. What are the two heat sinks in a heat engine?

The two heat sinks in a heat engine are the hot reservoir and the cold reservoir. The hot reservoir is the source of high temperature heat, while the cold reservoir is the destination for the released heat at a lower temperature.

4. What is the efficiency of a heat engine?

The efficiency of a heat engine is the ratio of the work output to the heat input. It is a measure of how well the engine converts heat into work. The maximum efficiency for a heat engine is determined by the Carnot cycle and is dependent on the temperatures of the two heat sinks.

5. How is the efficiency of a heat engine improved?

The efficiency of a heat engine can be improved by increasing the temperature difference between the two heat sinks, using a more efficient working fluid, and minimizing friction and other energy losses within the engine.

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