Three identical rods are joined in series as shown. Their thermal conductivities are `K, K//2` and `K//3` respectively. At steady, if the free ends of rods are at `100^(@)C` and `20^(@)C` respectively. Determine the temperature at two junction points. Also find the equivalent thermal conductivity.

Explanation

Let temperature of junctions are `theta_(1)` and `theta_(2)` respectively
`R_(1) = (L)/(KA) = R_(0), R_(2) = (L)/((K)/(2) A) = 2 R_(0), R_(3) = (L)/((K)/(3) A)= 3 R_(0)`
Heat current is same in all rods
`i= (100 - theta_(1))/(R_(1)) = (theta_(1) - theta_(2))/(R_(2)) = (theta_(2) - 0)/(R_(3))`
`(100 - theta_(1))/(R_(0)) = (theta_(1) - theta_(2))/(2R_(0)) = (theta_(2))/(3 R_(0))`
`600 - 6 theta_(1) = 3 theta_(1) - 3 theta_(2) = 2 theta_(2)`
`3 theta_(1) - 3 theta_(2) = 2 theta_(2) implies 3 theta_(1) = 5 theta_(2) implies theta_(2) = (3 theta_(1))/(5)`
`600 - 6 theta_(1) = 3 theta_(1) - 3 theta_(2) implies 9 theta_(2) implies 9 theta_(1) - 3 theta_(2) = 600`
`3 theta_(1) - theta_(2) = 200 implies theta_(1) - (3 theta_(1))/(5) = 200`
`12 theta_(1) = 1000 implies theta_(1) = (250)/(3) .^(@)C`
`theta_(2) = (3 theta_(1))/(5) = 50^(@)C`
`R_(eq) = R_(1) + R_(2) + R_(3) = R_(0) + 2 R_(0) + 2 R_(0) + 3 R_(0) = 6 R_(0)`
`(3L)/(K_(eq) A) = (6L)/(KA) implies K_(eq) = (K)/(2)`