Heat Transfer Problem - Help Needed

[JD88]

Sorry for the long post but I could use some help.

I am having a problem with a professor in my heat transfer class and I was wondering if anyone could help. Before I go on, I am not looking to get points back on my test, I only lost 2 so I don't care, I just want to know whether I am right or not.

We recently had a test where there was a question in which we had to use the finite difference method to solve for the unknown temperatures in a rectangular 2D block.

The professor took the question from the textbook and modified it for the test. In the test the 2D block was insulated on two sides and the other two sides were held at 300 C. The known temperatures at various nodes inside the block were HOTTER than 300 C and there was NO heat being added to the system. The prompt told us the block was in a STEADY STATE condition and that we should us the finite difference method to solve for the temperatures at the various unknown nodes.

Now, my problem is that this situation cannot possibly be in a steady state condition. Two sides insulated, two sides held at 300 C and temperatures within being hotter than 300 C with NO heat addition. The heat from inside must be conducting too the the boundaries held at 300C until the entire block is at a uniform temperature of 300 C.

Now the issue here is that during this test nobody (including the professor) realized that this situation could not be at steady state and we all solved the problem using the steady state form of the heat diffusion equation with no heat generation. When we received our tests back nearly the entire class had gotten the problem wrong. The professor then showed us his method in which he was also used the steady state heat diffusion equation without heat generation and he arrived at a different solution than the rest of the class. He used the equation differently than the rest of us and arrived at a different answer however when I showed him my solution he was unable to explain why it was wrong and just assumed that I typed the numbers into my calculator wrong, which is ridiculous because the majority of a 30+ student class arrived at the same answer as me.

So I have told my professor that this question was flawed because it told us to assume steady state but it is clearly a transient problem. However he still believes that if you just assume steady state (even though it is clearly not) you will arrive at his answer.

So finally why I need your help. It is obvious to me that applying the steady state diffusion equation with no heat generation to this problem can lead to different (incorrect) solutions depending on the particular way you apply it. Is there a way to prove this to the professor other than just applying it in different ways and arriving at different solutions? Because if I just work out the problem in different ways and arrive at different answers he always assumes I made an error and he won't try it for himself.

 

[Mapes]

I concur that the only possible steady-state solution for the system is a uniform 300°C temperature. One point you might make is that if an internal node is at >300°C, then there must be a temperature gradient between that node and the 300°C boundary, and therefore thermal energy must be flowing down this gradient. This would require heat generation in a steady-state system; if heat generation is ruled out, we have a logical inconsistency.

If you post the professor's solution, it should be relatively easy to show specifically how conservation of energy isn't being satisfied.

 

[JD88]

Thanks, I will post it but later today.

 

[JD88]

Here is my professor's response when I explained that this problem was not in a steady state condition.

"The question set is a theoretical situation that IF the system was in equilibrium what would the temperatures have to be. It does not ask how it got there nor what will happen later. It could be achieved by shining a laser light onto T2 so that it was slowly heated and when it reached 421C it was turned off. The system would then be in equilibrium. At a later time this would not be true but the question does not cover that period."

"If you set up the system with the temperatures I derived in class then the system will be in equilibrium until conduction starts at the left hand boundary to conduct heat away. Thus there is a very short finite time that the system will be in quasi-equilibrium. It is the same idea as in thermodynamics of quasi equilibrium."

 

[Mapes]

I am sorry that you have this professor.

Here is my professor's response when I explained that this problem was not in a steady state condition.

"The question set is a theoretical situation that IF the system was in equilibrium what would the temperatures have to be. It does not ask how it got there nor what will happen later. It could be achieved by shining a laser light onto T2 so that it was slowly heated and when it reached 421C it was turned off.

Why didn't your professor simply formulate the problem so that node T2 is kept at 421°C via some mechanism such as a laser? Then the system would truly be in equilibrium and the whole class wouldn't have been confused.

"If you set up the system with the temperatures I derived in class then the system will be in equilibrium until conduction starts at the left hand boundary to conduct heat away.

This let's me know that this person is confused about heat transfer. If an interior node is at >300°C and the boundary condition fixed at 300°C, then conduction doesn't "start," it's already occurring. What they probably meant is that a temperature decrease initiates at the interior as soon as the heat generation mechanism disappears (and it's this intuited temperature change that triggered your discomfort).

Look on the positive side: students often experience greatest understanding when they detect and identify their instructors' inconsistencies.

 

[JD88]

Thanks!

I knew there was something wrong with his argument

"the system will be in equilibrium until conduction starts at the left hand boundary to conduct heat away"

But you make an excellent point that conduction doesn't start, its already occurring.

I can't say that I am definitely just going to drop this with the professor but at least I know that I understand what is going on.

 

[Mech_Engineer]

Well it seems to me that since he didn't knock many points off anyway, you might consider dropping the subject for a couple of reasons:

• Arguing tiny minutia with your prof. will get you nowhere. You have pointed out the problem, and the professor sees that you know it is there and understand why; this should be enough.
• Given the problem statement and the fact that there is a "discrepancy" in the nodal temperatures, you should probably just assume there is heat flowing into the system to achieve the steady-state condition that is stated. The wording of the problem was problematic (especially if it said there is no heat flow into the system) but the problem is obviously trivial if the whole block is at the same temperature for steady state. If the professor put the question in the test, it's likely he didn't want you to use the finite difference method on a block of uniform temperature, so the "no heat flow into the block" statement is probably erroneous.
• Sometimes, you have to just realize a professor wants his/her answer, and you can easily just give it to them and move on. Fighting with them over 2 lousy points will only end in tears (espacially if they get angry and end up grading you more harshly for the rest of the semester). My best advice is drop it and move on, going farther down this path will not end well.

 

[JD88]

Well it seems to me that since he didn't knock many points off anyway, you might consider dropping the subject for a couple of reasons:

• Arguing tiny minutia with your prof. will get you nowhere. You have pointed out the problem, and the professor sees that you know it is there and understand why; this should be enough.
• Given the problem statement and the fact that there is a "discrepancy" in the nodal temperatures, you should probably just assume there is heat flowing into the system to achieve the steady-state condition that is stated. The wording of the problem was problematic (especially if it said there is no heat flow into the system) but the problem is obviously trivial if the whole block is at the same temperature for steady state. If the professor put the question in the test, it's likely he didn't want you to use the finite difference method on a block of uniform temperature, so the "no heat flow into the block" statement is probably erroneous.
• Sometimes, you have to just realize a professor wants his/her answer, and you can easily just give it to them and move on. Fighting with them over 2 lousy points will only end in tears (espacially if they get angry and end up grading you more harshly for the rest of the semester). My best advice is drop it and move on, going farther down this path will not end well.

I would like to point out that the test was only out of 20 points. So 2 points actually amounts to entire letter grade. But regardless of that fact, I know that I am right and he is wrong and I know this subject well enough that this lousy test grade isn't going to keep me from getting an A so I am going to drop it. Though I don't know about the rest of the class, some people are still pretty pissed.

 

[Mech_Engineer]

I would like to point out that the test was only out of 20 points. So 2 points actually amounts to entire letter grade. But regardless of that fact, I know that I am right and he is wrong and I know this subject well enough that this lousy test grade isn't going to keep me from getting an A so I am going to drop it.

Good choice, I knew a few people in college that just didn't know when to quit arging with the professor, and ended up deeper in the hole than when they started. Consider this a lesson in real-life politics, because you will find similar situations in day-to-day life at your job as well.

Though I don't know about the rest of the class, some people are still pretty pissed.

Let them worry about it, it really isn't worth getting that pissed over IMO. Life (and college) is too short to worry about one poorly worded problem on a test.