SC83
Mechanical
- Jun 8, 2018
- 24
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:
[ul]
[li]The temperature through the rod goes from 300 K to 20 K.[/li]
[li]The rod has a reduced cross-section A2 << A1. A2 has a length L2. The cross-section A1 = A3.[/li]
[li]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.[/li]
[li]The thermal conductivity changes with the temperature, i.e. k(T) = k0 + aT.[/li]
[/ul]
Question: why is Q not changing with L1 and L3 and how can I prove this mathematically (see attachment Link)?
Boundary conditions:
[ul]
[li]The temperature through the rod goes from 300 K to 20 K.[/li]
[li]The rod has a reduced cross-section A2 << A1. A2 has a length L2. The cross-section A1 = A3.[/li]
[li]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.[/li]
[li]The thermal conductivity changes with the temperature, i.e. k(T) = k0 + aT.[/li]
[/ul]
Question: why is Q not changing with L1 and L3 and how can I prove this mathematically (see attachment Link)?