Heat Conduction of Metal Rod (Differential Equations)

In summary, the student is having difficulty understanding how to solve for the steady state temperature in a copper rod with a heat source at one end and a reservoir at a different end. He initially uses a differential equation to find the steady state temperature, but later realizes he is not solving for the correct equation. He is eventually able to use more complicated analysis to find the steady state temperature.
  • #1
gohoubi
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Hi everyone, I recently started studying heat conduction using differential equations and this has been stumping me for a while.

I am having trouble understanding what type of heat conduction problem this is.

We are given a 100cm long copper rod with ends maintained at 0 C. The center of the bar is heated to 100 C by an external heat source and this is maintained until a steady state temperature is reached.

Then at time t = 0, after the steady state has been reached, the heat source is removed. At the same instant let the end x = 0 be placed in thermal contact with a reservoir at 20 C while the other end remains at 0 C.

So first, I decided this problem appeared similar to other problems with steady state solutions so I set it up like this:

u(x,t) = v(x) + w(x,t)

The steady state solution demands that v''(x) = 0.

I solve and find that v(x) = 2x for 0<=x<=50 and 200-2x for 50<x<=100.

However, when I try to set up the problem like previous examples, I do not get a homogeneous equation for w(x,t), because the left end temperature of u(x,t) is different from what I would get from v(x) (the steady state solution).

I am thinking I am approaching this problem incorrectly.

EDIT: I see part of my problem now. What I solved up above was f(x) and not v(x). v(x) should be 20-x/5 correct?
 
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  • #2
gohoubi said:
let the end x = 0 be placed in thermal contact with a reservoir at 20 C while the other end remains at 0 C.

I don't quite understand your symbols, but yea, the steady state solution would have a linear temperature gradient of T(x) = 20-x/5 as you say.
 
  • #3
I don't know if you are an undergrad. or grad. student. But this problem is more like a problem where complicated mathematics comes into play.. but no worries. I will help walk through it.

Initially your copper rod is maintained at ends of 0, C and then at the center of the bar is heated to 100 C, until steady state temperature is reached. The whole way you can figure out the temperature distribution in the rod, which is crucial to the second part where the reservoir takes place is that you try to mathematically solve for the steady state solution.

Again, I don't know if you are an undergrad. or grad. But you must utilize the cylindrical version of the heat equation, see the first equation on this pdf

http://www.ewp.rpi.edu/hartford/~ernesto/S2004/CHT/Notes/s06.pdf

you can ignore thetha, azithumal directions, and just use this equation.

dT/dt=(alpha/r)*(d/dr)*((r*dT/dr))

for steady state dT/dt=0, so 0=(alpha/r)*(d/dr)*((r*dt/dr)) OR
0=d/dr((r*dT/dr)), you integrate two times in respect to r, and use that solution as well as the boundary conditions in respect to r(0)=0 C, r(50)=100 C, to find the expression of T(r) for steady state. These also form the T(0,r) initial conditions for the next solution.

now for the next part where there is a heat flux at the end, you need to use more complicated analysis.. but I'm not sure what they are asking for? you didn't specify?
 
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Likes Adel Makram

1. What is heat conduction and how does it relate to metal rods?

Heat conduction is the transfer of thermal energy from a higher temperature object to a lower temperature object through direct contact. In the case of metal rods, heat conduction occurs when thermal energy is transferred from one end of the rod to the other end due to the difference in temperature at each end.

2. How is heat conduction in metal rods described by differential equations?

Differential equations are used to describe the rate of change of a physical quantity over time. In the case of heat conduction in metal rods, the differential equation used is the heat conduction equation, which relates the rate of change of temperature with respect to time to the thermal conductivity, cross-sectional area, and length of the rod. This equation helps to model and predict the temperature distribution along the rod over time.

3. What factors affect the heat conduction of metal rods?

The three main factors that affect heat conduction in metal rods are thermal conductivity, cross-sectional area, and length of the rod. Generally, materials with higher thermal conductivity will conduct heat more easily, while a larger cross-sectional area and shorter length of the rod will result in faster heat conduction. Other factors that can affect heat conduction include temperature gradient, surface area, and the presence of any insulating materials.

4. How does heat conduction in metal rods differ from other materials?

Metal rods typically have a higher thermal conductivity compared to other materials, such as wood or plastic. This means that they can conduct heat more easily and efficiently. Additionally, the heat conduction equation for metal rods may differ from that of other materials due to differences in thermal properties, such as thermal conductivity and specific heat capacity.

5. Can heat conduction in metal rods be controlled?

Yes, heat conduction in metal rods can be controlled through various methods. One way is by changing the material of the rod to one with a lower thermal conductivity, which will result in slower heat conduction. Additionally, insulating materials can be used to reduce heat transfer through the rod. The length and cross-sectional area of the rod can also be manipulated to control the rate of heat conduction.

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