Question

Two rods of different nature of same cross-sectional area and lengths having thermal conductivity k and 2k are joined. What is the equivalent thermal conductivity of the composite rod?

Answer 1

No No never

Heat transfer depends on area of cross section and length. 
Say heat transfers lengthwise

So in first condition heat transfer would occur via cross section ie of circle and accross length 2l 
And in second case heat transfer would be through cross section of somewhat 8 type (two circle adjacent to each other) and across the length l.

Q = temp differnce/resistance
Resistance = L/AK
If temp diff be same then Q is proptional to A/L

So

Q1/Q2 = A1/A2 x L2/L1
D1 is D and A2 would be 2xA1

So Q1/Q2 = 1/4