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In fig. 11.13 two metal bars of thicknes...

In fig. 11.13 two metal bars of thickness `x_1 and x_2` and coefficient of thermal conductivities `K_1 and K_2`, placed end to end. What is the coefficient of thermal conductivity of a uniform slab of thickness `x_1 + x_2` whose conductivity is the same as that of the system of two bars ? Assume that there is no heat loss from the lateral sides.

Text Solution

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When the system attains steady state,
`K_1 = ((T_1 - T)/( x_1 )) = K_2 ((T- T_2)/( x_1 )) = (K (T_1 - T_2 ))/( x_1 + x_2) " "…(i)`
From problem 11.15, we have
`T = ((K_1 T_1)/( x_1 ) + (K_2 T_2)/( x_2) )/((K_1)/( x_1) + (K_2)/(x_2) )`
substituting in equation (i)
`(K_1 )/( x_1 ) [ ((T_1 K_1)/( x_1) + (T_1 K_2)/( x_1) - (K_1 T_1 )/(x_1) - (K_2 T_2)/(x_2) )/( (K_1)/(x_1) + (K_2)/(x_2) ) ] = K(T_1 - T_2)/( x_1 + x_2)`
`K= (k_1 k_2 ((x_1 + x_2) )/( x_1 x_2) )/( (K_1)/(x_1) + (K_2)/(x_2) ) , K = (x_1 + x_2)/( (x_1)/(K_1) + (x_2)/(K_2) )`
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