Heat Conduction of Metal Rod (Differential Equations)

[gohoubi]

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?

 

[Unrest]

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.

 

[yus310]

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?