Three metal rod A, B and C of same length and cross-section are placed end to end and a temperature difference is maintained between the free ends of A and C. If the thermal conductivity of `B(K_B)` is twice that of `C(K_C)` and half that of `A(K_A)`, then the effective thermal conductivity of the system will be

Options

A. `(K_(X))/(7)`
B. `(6K_(Y))/(7)`
C. `(7K_(Y))/(3)`
D. `7K_(Z)`

Answer

B

Explanation

Given `K_(Y) = 2K_(Z), K_(Y) = (1)/(2) K_(X)`
Let L and A be the length and area of cross-section of each rod respectively. Then
Thermal resistance of rod X, `R_(X) = (L)/(K_(X)A)`
Thermal resistance of rod Y, `R_(Y) = (L)/(K_(Y)A)`
Thermal resistance of rod Z, `R_(Z) = (L)/(K_(Z)A)`
The electrical equivalent circuit is as shown in the figure.

Here, the rods are connected in series
`:. R_(S) =R_(X) + R_(Y) + R_(Z)`
or `(3l)/(K_(eq)A) = (L)/(K_(X)A) + (L)/(K_(Y)A) + (L)/(K_(Z)A)`
or `(3)/(K_(eq)) = (1)/(2K_(Y)) + (1)/(K_(Y)) + (2)/(K_(Y)) = (1 + 2 + 4)/(2K_(Y)) = (7)/(2K_(Y))`
or `K_(eq) = (6)/(7) K_(Y)`