Fourier's Equations: difficult problem

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SUMMARY

The discussion revolves around solving a heat transfer problem involving Fourier's equations, specifically calculating the heat transfer rate (Q/t) in a system with varying thermal conductivity (kappa) due to temperature differences. The user is tasked with analyzing a series of cylindrical and parallel insulators transitioning from an initial temperature (Tzero) to a final temperature (Ttwo), with kappa assumed to be a linear function of temperature. The user has attempted to separate and integrate the equations but struggles with establishing a general form for the thermal resistance (R) and determining whether the scenario represents a transient or steady-state problem.

PREREQUISITES
  • Understanding of Fourier's Law of Heat Conduction
  • Familiarity with thermal conductivity and its temperature dependence
  • Basic knowledge of calculus for integration and separation of variables
  • Concepts of transient and steady-state heat transfer analysis
NEXT STEPS
  • Study the derivation and application of Fourier's Law in varying thermal conductivity scenarios
  • Learn about transient heat conduction and its mathematical modeling techniques
  • Explore the concept of thermal resistance in series and parallel configurations
  • Investigate graphical methods for visualizing heat transfer problems
USEFUL FOR

Students and professionals in thermal engineering, mechanical engineering, and physics who are dealing with heat transfer problems, particularly those involving varying thermal properties in complex geometries.

argpirate
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Homework Statement


So I have a problem where I have a series and two other parallel insulators bringing a system from a temperature of Tzero to Ttwo. However, the two aforementioned temperatures have a huge difference thus causing kappa to vary with temperature. I am trying to know how to solve for Q/t of the whole system accounting for the varying kappa with temperature (assume the kappa function is linear with temperature change (y=mx+b form)). So I have a cylinder that is in series (2*pi*r*L) and top and bottom that are parallel but not series (disks with thickness).

Homework Equations


Q/t=kAdT/dx

The Attempt at a Solution


I serperated and integrated. I set Q/t=Q/t for the series from where they change. The difficulty I have is coming up with the general form (T-t)/R and using that to do stuff.
 
Physics news on Phys.org
Is this a transient problem or a steady state problem?
 
I think we need a picture ...
 

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