- #31
member 428835
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?Chestermiller said:
Heat Equation for Cylinder Wire Problem
- Context: Graduate
- Thread starter member 428835
- Start date
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- Tags
- Cylinder Heat Heat equation Wire
Click For Summary
Discussion Overview
The discussion revolves around the application of the heat equation to a cylindrical wire problem, focusing on the heat generated due to electrical resistance and the subsequent temperature distribution within the wire. Participants explore the mathematical formulation of the heat equation, the assumptions involved, and the implications of different parameters such as radius and thermal conductivity.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant presents the heat produced in the wire as $$Q = R I^2 \pi r_i^2 L$$ and questions whether $$Q$$ should be divided by an arbitrary radius $$r$$ or the wire's radius $$r_i$$.
- Another participant emphasizes the need for units to be consistent, noting that heat generation in three dimensions should be in watts per cubic meter.
- There is a discussion about the units of $$k \nabla^2 T$$, with participants clarifying that $$k$$ refers to thermal conductivity.
- Participants debate the correct formulation of $$Q$$ and its dependence on the radius, with some suggesting that $$Q$$ should be defined as $$Q = I^2 R / \pi r_i^2 L$$.
- There is a proposal to integrate the heat equation from 0 to $$r_i$$, with discussions on the bounds of integration and whether to consider a profile or a specific value.
- One participant expresses confusion over differing results from a flux balance and the heat equation method, leading to a request for clarification on the algebra involved.
- Participants explore the implications of uniform heat generation within the wire and how it affects the heat generation rate per unit volume.
- There is a challenge to reconcile the heat generation expressions derived from different approaches, with a focus on understanding the physical meaning behind the equations.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the correct formulation of the heat equation or the interpretation of the results. Multiple competing views remain regarding the integration bounds and the relationship between heat generation and flux balance.
Contextual Notes
There are unresolved questions about the assumptions made regarding the uniformity of heat generation and the implications of using different radii in the equations. The discussion also highlights potential discrepancies in the mathematical steps taken by participants.
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