Proving a thermal conductivity/resistance formula.

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SUMMARY

The discussion centers on deriving the temperature at the interface of a block composed of two materials with distinct thermal conductivities (λ1, λ2) and thicknesses (l1, l2). The formula for the interface temperature (Tx) is established as Tx = (R1 * Thigh + R2 * Tlow) / (R1 + R2), where R1 and R2 represent the thermal resistances calculated using R = l/λ. This approach parallels the analysis of electrical resistors in series, emphasizing the relationship between heat flow and temperature differences.

PREREQUISITES
  • Understanding of thermal resistance equations (R = l/λ)
  • Knowledge of heat transfer principles
  • Familiarity with series circuits in electrical engineering
  • Basic algebra for manipulating equations
NEXT STEPS
  • Study the derivation of thermal resistance in composite materials
  • Learn about heat transfer mechanisms in solids
  • Explore the analogy between thermal and electrical circuits
  • Investigate the impact of material properties on thermal conductivity
USEFUL FOR

Students in physics or engineering disciplines, particularly those focusing on thermodynamics and heat transfer, will benefit from this discussion.

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Homework Statement


The figure below shows a block that consists of two materials with different thicknesses l 1 and l 2 and different thermal conductivities λ1 and λ2. The temperatures of the outer surfaces of the block are Thigh and Tlow, as shown in the figure. Each face of the block has a cross-sectional area A. Using the thermal resistance equation R = l/λ,show that the temperature Tx at the interface of the block is given by:
Tx = R1Thigh + R2Tlow/R1 + R2


Homework Equations


R=l/λ


The Attempt at a Solution


I'm not even sure where to begin. A nudge in the right direction would be greatly appreciated.
 

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You will need a separate equation for the heat flow through each block. (I assume you have an equation relating heat flow, temperature difference etc.) Invent unknowns as necessary. Some unknowns will be common to the two equations.
 
Think about two electrical resistors in series between a voltage of Vhigh and Vlow. Get the voltage at the junction of the two resistors.
 

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