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Question

What is equivalent thermal conductivity of a compound rod? (and its derivation/formula)

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Solution

Dear Student,

Heat Transfer( heat current ) Q˙ is directly proportional to the thermal conductivity of a material ( it is specific to a material ), drawing analogy from current flowing through different conductors with different current condivity we can deduce that the flow of heat through a compound system ( made with more than one material ) will be different and will depend on the effective conductivity co-efficient of the system ( same as that in current electricity )

using this knowledge and the fact

Q˙=-KAT'-TL
we define our thermal resistance,

Q˙=T-T'LKAdrawing analogy from I=VRwe define thermal resistance Rth=LKAwhere L=length of conductor, A= area of cross section and K its thermal conductivity
Now there can be two situations, conductors placed end-end(series ) or one on top of another ( parallel )

(a) Series

RThseries=Rth1+Rth2 ( same as current electricity resistance in series )L'KeqA'=L1K1A1+L2K2A2lets assume L1=L2=L thus L'=2L and A1=A2=A thus A'=A2LKeqA=LK1A+LK2A2Keq=1K1+1K2or, Keq=2K1K2K1+K2

(b) parallel

1Rthparallel=1Rth1+1Rth2KeqA'L'=K1A1L1+K2A2L2 for L1=L2=L and A1=A2=A we haveKeq2AL=K1AL+K2ALor, Keq=K1+K22

Regards.



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