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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?
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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