First slide

Heat transfer

Question

Three rods of same dimensions have thermal conductivities 3K, 2K and K, they are arranged as shown in the figure, then the temperature of the junction (T) in steady state is

Moderate

A

$\frac{100^{0}}{3} C$
B

$\frac{200^{0}}{3} C$
C

$75^{\circ} c$
D

$\frac{50}{3}^{\circ} c$

Solution

$\begin{array}{rcl} a m o u n t o f h e a t f l o w i n g \\ f r o m l e f t t o r i g h t a t t h e j u n c t i o n i s d i s t r i b u t e d b e t w e e n t w o r o d s \\ a m o u n t o f h e a t f l o w i n f i r s t r o d i s Q & = & \frac{2 K (A) (T - 50) t}{l} + \frac{K A (T - 0) t}{l} = \frac{3 K (A) (100 - T) t}{l} - - - - (1) \\ s o T & = & \frac{200^{0} c}{3} \\ Q h e a t i s d i s t r i b u t e d b y t h e o t h e r t w o r o d s \\ k i s c o e f f i c i e n t o f t h e r m a l c o n d u c t i v i t y \\ θ_{2}, θ_{1} a r e t e m p e r a t u r e s \\ t i s t i m e \\ l i s l e n g t h o f t h e r o d \end{array}$

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