Question 577: Thermodynamics & KTG

Question

Three conductors of same length having thermal conductivity $k_1, k_2$ and $k_3$ are connected as shown in figure.

Area of cross sections of $1^{	ext {st }}$ and $2^{	ext {nd }}$ conductor are same and for $3^{	ext {rd }}$ conductor it is double of the $1^{	ext {st }}$ conductor. The temperatures are given in the figure. In steady state condition, the value of $	heta$ is
_________ ${ }^{\circ} \mathrm{C}$.
(Given : $\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$ )

Accepted Answer

$${R_1} = {{2L} \over {{K_1}A}}$$
(As $${l_1} = {l_2} = {l_3} = l$$
$${A_1} = {A_2} = A/2$$
$${A_3} = A$$)
$${R_2} = {{2L} \over {{K_2}A}}$$
$${R_3} = {L \over {{K_3}A}}$$
$${I_1} + {I_2} = {I_3}$$
$$ \Rightarrow {{100 - 	heta } \over {{{2L} \over {{K_1}A}}}} + {{100 - 	heta } \over {{{2L} \over {{K_2}A}}}} = {{	heta  - 0} \over {{L \over {{K_3}A}}}}$$
$$ \Rightarrow {{100 - 	heta } \over 2}({K_1} + {K_2}) = {K_3}	heta $$
$$ \Rightarrow 50({K_1} + {K_2}) = 	heta \left[ {{K_3} + {{{K_1} + {K_2}} \over 2}} ight]$$
$$ \Rightarrow 50(60 + 120) = 	heta \left[ {135 + {{60 + 120} \over 2}} ight]$$
$$ \Rightarrow 50 	imes 80 = 	heta (135 + 90)$$
$$ \Rightarrow 	heta  = {{50 	imes 180} \over {225}}$$
$$ \Rightarrow 	heta  = 40^\circ C$$

Thermodynamics & KTG

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