First slide

Heat transfer

Question

Consider a compound slab consisting of two different materials having equal thickness and thermal conductivities K and 2K respectively. The equivalent thermal conductivity of the slabs taken in series is

Moderate

A

$\frac{2}{5} K$
B

$3K$
C

$\sqrt{2} K$
D

$\frac{4}{3} K$

Solution

$r e s i s \tan c e i n s e r i e s i s R_{s e r i e s} = R_{1} + R_{2} w h e r e R = \frac{l}{K A}, h e r e l i s t h i c k n e s s o r l e n g t h, A i s a r e a o f c r o s s s e c t i o n, K i s c o e f f i c i e n t o f t h e r m a l c o n d u c t i v i t y w h e n 2 s l a b s o f d i f f e r e n t c o e f f i c i e n t o f t h e r m a l c o n d u c t i v i t y K, 2 K, h a v i n g s a m e t h i c k n e s s a n d a r e a o f c r o s s s e c t i o n a r e t a k e n i n s e r i e s e f f e c t i v e t h e r m a l c o n d u c t i v i t y i s K_{s e r i e s}, g i v e n b y, \frac{l + l}{K_{s e r i e s}} = \frac{l}{K} + \frac{l}{2K} \Rightarrow \frac{2}{K_{s e r i e s}} = \frac{3}{2 K} \Rightarrow K_{s e r i e s} = \frac{4}{3} K$

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