Question

Two slabs of thermal conductivity K₁and K₂, length l₁, and l₂ of the same cross section are placed in series. Show that the series combination of two slabs behaves as a single slab of thermal conductivity K given by

$space space space space space space space space space straight K equals fraction numerator straight l subscript 1 plus straight l subscript 2 over denominator begin display style straight l subscript 1 over straight K subscript 1 plus straight l subscript 2 over straight K subscript 2 end style end fraction$

Solution

Let A be the area of the cross-section of each slab.

Thermal resistance of slab of thermal conductivity K₁ is,

straight R subscript 1 equals fraction numerator straight l subscript 1 over denominator straight K subscript 1 straight A end fraction space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis space

Thermal resistance of slab of thermal conductivity K₂ is,

space space space space space R equals fraction numerator l subscript 1 plus l subscript 2 over denominator K A end fraction space space space space space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis space

For series combination,

space space straight R equals straight R subscript 1 plus straight R subscript 2 space space rightwards double arrow space space space fraction numerator straight l subscript 1 plus straight l subscript 2 over denominator K A end fraction equals fraction numerator straight l subscript 1 over denominator straight K subscript 1 straight A end fraction plus fraction numerator straight l subscript 2 over denominator straight K subscript 2 straight A end fraction space space rightwards double arrow space space space space space space space space space space space space space straight K equals fraction numerator straight l subscript 1 plus straight l subscript 2 over denominator begin display style straight l subscript 1 over straight K subscript 1 plus straight l subscript 2 over straight K subscript 2 end style end fraction space

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