Thermodynamics problem, Heat Engine with two heat sinks

[TeddyLu]

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:

T_H = 1000 K
T_C1 = 200 K
T_C2 = 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 - T_C/T_H = 1 - Q_C/Q_H

The Attempt at a Solution

Since there was two heat sinks with an equal value of heat rejected at each:
Q_C1 = Q_C2 = Q_C

therefore, Q_H = Q_C1 + Q_C2 will turn into
Q_H = 2Q_C

I took the entropy balance equation to solve for Q_C:

0 = Q_H/1000 - Q_C/200 - Q_C/300

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

any help please?

 

[siddharth23]

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.

 

[rude man]

therefore, Q_H = Q_C1 + Q_C2 will turn into
Q_H = 2Q_C

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.

 

[TeddyLu]

@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 (Q_C1 & Q_C2) and solved for Q_C/Q_H to plug back into my equation for the carnot efficiency.

Appreciate all the help for this problem!

 

[rude man]

@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 (Q_C1 & Q_C2) and solved for Q_C/Q_H to plug back into my equation for the carnot efficiency.

Appreciate all the help for this problem!

Good. So what was your answer?