What Is the Temperature Distribution Along an Insulated Rod?

[stripes]

Homework Statement

A very simple problem, but part of an assignment regarding heat flow. The question is:

A 15 cm rod is insulated at one end, the other is in contact with a reservoir of temperature 37C. By thinking of the physics, after a long time, what is the temperature at the point 5 cm and 10 cm from the insulated end?

Homework Equations

None

The Attempt at a Solution

So I will say that I was sick for a while and missed a week of class. I've been catching up so I'm somewhat familiar with things. Obviously this question is very simple, but I have been thinking of the following questions:

What does "insulated at one end" mean? It is 0 C? Does the reservoir stay at 37C forever? If so, then after a long time, the entire rod would be 37C. This is probably not the case. If the reservoir transfers and loses it's heat, then after a long time, the rod will reach a uniform, final temperature. This, however, depends on the temperature at the other "insulated" end, doesn't it? Or would the temperature distribution be 12.333 C at 5 cm, 24.666 C at 10 cm, and 37 C at 15 cm?

Basically, since I missed some class, I am unfamiliar with some of the basic ideas that most students would have been informed about in class. So how do I figure out (by "thinking of the physics" as opposed to calculating) what the temperature will be at various points, given the information above?

Thanks in advance for any help.

 

[Saladsamurai]

Hi stripes

Generally, in thermodynamic problems the term "reservoir" does in fact mean that it stays the same temperature for ever.

Why should "insulated" imply some specific temperature (zero as you suggested)? Are you familiar with the term "adiabatic"? What does it mean?

 

[stripes]

I just took a guess, I wasn't sure. But of course none of the heat will leave the other end of the rod.

So given that the reservoir stays at 37C forever, the rod would eventually be 37C at every point, correct? That's just how it works, isn't it? Just like when the end of time comes, the universe will cool to 2K or whatever the figure is.

 

[Chestermiller]

I don't know about the universe, but your analysis of the rod problem is spot on.

 

[paranormal]

I would suggest starting by reviewing the basic principles of heat transfer and thermal equilibrium. The concept of insulation means that there is no heat exchange between the insulated end of the rod and its surroundings. This means that the temperature at this end will not change, and it is not necessarily 0 C. The reservoir may also not stay at 37C forever, as it may lose heat over time depending on its surroundings.

To determine the temperature at various points along the rod, we can use the principle of thermal equilibrium, which states that in a closed system, heat will flow from a higher temperature to a lower temperature until both reach the same temperature. In this case, the reservoir is the source of heat and the rod is the system.

After a long time, the rod will reach a uniform, final temperature. This temperature will depend on the thermal conductivity of the rod material, the initial temperature of the rod, the temperature of the reservoir, and the distance from the insulated end. You can use the equation for one-dimensional heat conduction to calculate the final temperature at each point along the rod.

I would also recommend discussing this problem with your instructor or peers to gain a better understanding of the concepts and principles involved. It may also be helpful to review any notes or materials from the class that you missed. Good luck with your assignment!