Heat Transfer Between Two Rods in Contact

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SUMMARY

The discussion focuses on determining the equilibrium temperature distribution in two welded homogeneous rods with differing heat conductivities, \(\kappa_1\) and \(\kappa_2\), while maintaining continuity in temperature and heat flux at the weld. The governing equation for heat transfer, \(q_x = -kA\frac{dT}{dx}\), is essential for solving the problem. The specific heat \(c\) and density \(\rho\) are relevant for calculating the diffusion constants \(k_j = \frac{\kappa_j}{c\rho}\), which influence the heat conduction behavior in the rods. The solution approach involves equating heat fluxes to find the temperature \(T_2\) at the interface.

PREREQUISITES
  • Understanding of heat transfer principles, specifically conduction.
  • Familiarity with the concepts of thermal conductivity and diffusion constants.
  • Knowledge of differential equations as applied to physical systems.
  • Basic grasp of temperature distribution in composite materials.
NEXT STEPS
  • Study the derivation of the heat conduction equation in one-dimensional systems.
  • Learn about the method of separation of variables for solving heat transfer problems.
  • Research the impact of boundary conditions on temperature distribution in composite materials.
  • Explore numerical methods for solving heat transfer equations, such as finite difference methods.
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Students and professionals in mechanical engineering, physics, and materials science who are dealing with heat transfer analysis, particularly in composite materials and thermal management applications.

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1.Two homogeneous rods have the same cross section, specific heat c, and density \rho but different heat conductivities \kappa1 and \kappa2. Let kj=\kappaj/(c\rho) be their diffusion constants. They are welded together so that the temperature u and the heat flux \kappaux at the weld are continuous. The left-hand rod has its left end maintained at temperature zero. The right-hand rod has its right end maintained at temperature T degrees.

Find the equilibrium temperature distribution in the composite rod.



Homework Equations


qx=-kA\frac{dT}{dx}


The Attempt at a Solution


Solution is in attached pdf with picture because it was easier to put the equations together using an external tool.

Equilibrium temperature distribution just means find the governing equation for the temperature of the rod, right?If that's the case then I just equate both q's and solve for T2. Another question I had about this problem is the specific heat c. Where does this come into play for solving this problem? Thanks for any help with this topic. It's been a little while since I've done any heat transfer stuff.
 

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