Heat conduction with variable conductivity
Heat conduction with variable conductivity
(OP)
I want to reduce the heat conduction through a steel rod by locally reducing the cross-section of the steel rod. However, it seems that the conduction through the rod is not dependent on the position of the reduced cross-section (i.e. near the hot or cold surface). I'm trying to explain this mathematically (see attachment Link), but I'm stuck. Can somebody help me?
Boundary conditions:
Question: why is Q not changing with L1 and L3 and how can I prove this mathematically (see attachment Link)?
Boundary conditions:
- The temperature through the rod goes from 300 K to 20 K.
- The rod has a reduced cross-section A2 << A1. A2 has a length L2. The cross-section A1 = A3.
- The total length of the rod doesn't change, i.e. L1 + L2 + L3 = L. The length of L2 is constant, L1 and L2 can vary to position the reduced cross-section near 300 K or 20 K.
- The thermal conductivity changes with the temperature, i.e. k(T) = k0 + aT.
Question: why is Q not changing with L1 and L3 and how can I prove this mathematically (see attachment Link)?
RE: Heat conduction with variable conductivity
From the simple perspective of Q = T * S, where S is the thermal conductivity in W/K, the equation doesn't care how you got the conductivity to go down.
TTFN (ta ta for now)
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RE: Heat conduction with variable conductivity
RE: Heat conduction with variable conductivity
By the way, it is not only true that Q is independent of L1 if k is linear with temperature. It's true for any arbitrary function k(T), even a discontinuous one. You can demonstrate this by numerically integrating Fourier's law for any k(T).
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