Nonhomogeneous BC Heat Conduction

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Discussion Overview

The discussion revolves around solving a heat conduction problem in a long cylinder with nonhomogeneous boundary conditions, specifically addressing the temperature distribution T(r, θ) when different temperatures are applied to different halves of the cylinder's surface. The focus is on the formulation of the governing equations and the appropriate boundary conditions in cylindrical coordinates.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents a heat conduction problem involving a cylinder with temperatures T1 and T2 applied to its surface and expresses confusion about defining the boundary conditions.
  • Another participant questions the inclusion of the z term in the Laplace equation, suggesting that a θ term should be present instead, indicating a potential misunderstanding of the problem's dimensionality.
  • A participant expresses their confusion regarding cylindrical coordinates and acknowledges their ability to solve the problem but struggles with formulating it correctly due to the inclusion of the z term.
  • A repeated expression of confusion about cylindrical coordinates leads to a request for clarification on the correct form of the Laplace equation that includes θ dependence while omitting z dependence.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the correct formulation of the governing equations, particularly regarding the inclusion of the z term versus the θ term in the Laplace equation. Multiple competing views and uncertainties remain regarding the proper approach to the problem.

Contextual Notes

There is a lack of clarity regarding the assumptions made in the problem formulation, particularly in the context of cylindrical coordinates and the dimensionality of the governing equations. The discussion reflects unresolved mathematical steps and potential misinterpretations of the boundary conditions.

obscure
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Homework Statement


A very long cylinder has temperature T1 impressed on half of its peripheral surface and
temperature T2 impressed on the other half. Find T(r, θ).

Homework Equations


governing eq 1/r ∂/∂r(r∂/∂r)+∂2T/∂z2

The Attempt at a Solution


I am right at r=0 ;T=T1 at r=r' T=T2
to make BC homog define new term Q=T-T1

I am confused while define BCs
 

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If T is a function of r and θ, why do you have the z term in your Laplace equation. You should have the θ term, not the z term.
 
I am always confused with cylindrical coordinates. I know how to solve this but I couldn't formulate the problem as you sad I added z term.
 
obscure said:
I am always confused with cylindrical coordinates. I know how to solve this but I couldn't formulate the problem as you sad I added z term.
So what is the laplace equation when you include the theta dependence and omit the z dependence?
 

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