One-dimensional steady state conduction in Cylindrical coordinates

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Discussion Overview

The discussion revolves around a homework problem involving one-dimensional steady state heat conduction in cylindrical coordinates. Participants are addressing the mathematical formulation of the problem, the application of boundary conditions, and the physical implications of the solutions derived from the governing equations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents a heat conduction problem involving a cylindrical fuel with specific dimensions and thermal properties, seeking to determine the temperature at the axis.
  • The participant attempts to use a differential equation for heat conduction but encounters difficulties with the logarithmic term when applying boundary conditions.
  • Another participant asserts that the constant c1 must be set to zero to ensure that the temperature remains finite at r = 0, as the natural logarithm diverges at that point.
  • A further response clarifies that rejecting the solution involving ln(r) is common practice in physics when the solution, while mathematically valid, does not have a physical interpretation.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of ensuring that the temperature remains finite everywhere, leading to the conclusion that c1 must be set to zero. However, there is some discussion regarding the reasoning behind this decision and its implications.

Contextual Notes

The discussion highlights the importance of physical validity in mathematical solutions, particularly in the context of boundary conditions. The implications of setting c1 to zero are not fully resolved, as participants explore the relationship between mathematical solutions and physical constraints.

Hannibal247
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Hello,
Im having some issues with my task.

1. Homework Statement
The heat generation rate of a cylindrical fuel (D=0.2 m and 1 m long) is 160 kW.
The thermal conductivity of the fuel is 100 W/mK and its surface temperature is
maintained at 283 K. Determine the temperature at the axis.

Homework Equations


I tried to use this equation: 0=(1/r)*(d/dr)*(r*dT/dr)+q/k and I've added the volume to it
--> 0=(1/r)*(d/dr)*(r*dT/dr)+q/(k*V)[/B]

The Attempt at a Solution


im getting this at the end: T(r)=-1/4*q/(kV)*r2 + c1*ln(r) + c2
i wanted to use the boundary condition that for r=0 ---> T=283K, but i can't type ln(0) into my calculator.
I don't know how to go on. Best regards
 
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Temperature has to be finite everywhere. Therefore c1 = 0
 
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Henryk said:
Temperature has to be finite everywhere. Therefore c1 = 0
ok thank you very much. i understand why the temperature has to be everywhere finite, but why does it make c1=0? how is the relation to that?
best regards
 
because ln(r) diverges at r = 0, that's you have to reject this solution, i.e. set c1 to zero. It does satisfy the differential equation but it is not physical. It is actually very common practice in physics to reject solutions that satisfy mathematics but are not physically correct
 
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