Heat Equation for Cylinder Wire Problem

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SUMMARY

The forum discussion revolves around the heat equation applied to a cylindrical wire with radius r_i, length L, resistance R, and current I. The heat produced is expressed as Q = R I^2 \pi r_i^2 L, and the participants debate the correct application of Fourier's law and the divergence theorem in this context. They clarify that the rate of heat generation per unit volume is constant and should be modeled correctly in the equations, particularly emphasizing that Q should be defined as Q = I^2 R / \pi r_i^2 L. The discussion concludes with insights on integrating the heat equation and the implications of radial positions.

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  • Understanding of the heat equation in cylindrical coordinates
  • Familiarity with Fourier's law of heat conduction
  • Knowledge of the divergence theorem in vector calculus
  • Basic principles of electrical resistance and power calculations
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  • #31
Chestermiller said:
The flux balance is wrong because it implicitly assumes that all the heat generation takes place between r = 0 and radial location r, and none of the heat is generated between r and ri.

Chet
ahh yes, this makes sense! so, if we were to look at the flux at some ##r > r_i## would we be able to use the flux argument?
 
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  • #32
joshmccraney said:
ahh yes, this makes sense! so, if we were to look at the flux at some ##r > r_i## would we be able to use the flux argument?
r > r1 is outside the wire. We don't know what's happening out there, do we?

Chet
 
  • #33
Chestermiller said:
r > r1 is outside the wire. We don't know what's happening out there, do we?

Chet
sorry, I'm speaking in hypotheticals. and yea, if it was the same material but no heat generation.
 
  • #34
joshmccraney said:
sorry, I'm speaking in hypotheticals. and yea, if it was the same material but no heat generation.
Then it would be OK.

Chet
 
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  • #35
Chestermiller said:
Then it would be OK.

Chet
Thanks chet!
 

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