A cylinder of radius r made of a material of thermal conductivity `K_1` is surrounded by a cylindrical shell of inner radius r and outer radius 2r made of a material of thermal conductivity `K_2`. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. Show that the effective thermal conductivity of the system is `(K_1 + 3K_2 )//4`.

Explanation

To solve the problem, we need to find the effective thermal conductivity of a system consisting of a solid cylinder surrounded by a cylindrical shell. Here’s the step-by-step solution:

### Step 1: Understand the System
We have:
- A solid cylinder of radius \( r \) with thermal conductivity \( K_1 \).
- A cylindrical shell surrounding it with inner radius \( r \) and outer radius \( 2r \), made of a material with thermal conductivity \( K_2 \).
- The length of both the cylinder and the shell is \( L \).
- The system is in a steady state with no heat loss across the cylindrical surface.
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