Heat Conduction with Insulated Ends

In summary, the problem involves finding the steady-state temperature distribution of a copper rod with a length of 100 cm, where the ends are maintained at 0 degrees C and the center is heated to 100 degrees C. This can be solved using the heat equation for steady state and applying the given temperature boundary conditions.
  • #1
k.kuhlmann
1
0

Homework Statement


Let the ends of a copper rod 100 cm long be maintained at 0 degrees C. Suppose that the center of the bar is heated to 100 degrees C by an external heat source and that this situation is maintained until a steady state results. Find this steady-state temperature distribution.


Homework Equations


I think this involves the Fourier Series but I'm not sure.


The Attempt at a Solution


I'm not sure how to start it. I haven't seen a problem like this where it is heated from the middle.
 
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  • #2
Hi k.kuhlmann, welcome to PF!

One way to approach this problem is to use the heat equation for steady state

[tex]\frac{d^2T}{dx^2}=0[/tex]

and solve it (integrate it twice, for example) at either the left or right side of the rod (they're reflectively symmetric, of course) with the temperature boundary conditions that you're given. Does this make sense?
 

1. What is heat conduction with insulated ends?

Heat conduction with insulated ends is the transfer of heat through a material, such as a solid or a fluid, without any external heat source. This process occurs when one part of the material is at a higher temperature than another, causing the heat to flow from the hotter region to the colder region.

2. How does insulation affect heat conduction?

Insulation reduces heat conduction by creating a barrier between the two regions of different temperatures. This barrier prevents the transfer of heat and slows down the rate of heat conduction. Insulation materials, such as foam, wool, or air pockets, are poor conductors of heat and therefore, limit the flow of heat between the two regions.

3. What are the applications of heat conduction with insulated ends?

Heat conduction with insulated ends has various applications in everyday life, such as in the design of building materials, cooking processes, and refrigeration systems. It is also used in industrial processes, such as heating and cooling systems, to control the transfer of heat and maintain a desired temperature.

4. What factors affect heat conduction with insulated ends?

The rate of heat conduction with insulated ends depends on several factors, including the temperature difference between the two regions, the thickness and type of insulation material, and the thermal conductivity of the material through which the heat is being transferred. Additionally, factors such as humidity, air movement, and surface conditions can also affect the rate of heat conduction.

5. How is heat conduction with insulated ends calculated?

The rate of heat conduction with insulated ends can be calculated using the thermal conductivity of the material, the distance between the two regions, and the temperature difference between the two regions. The formula used to calculate heat conduction is Q = (kAΔT)/d, where Q is the rate of heat conduction, k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and d is the distance between the two regions.

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