1. Example-cylindrical coordinates
Use the conduction equation to find an expression for the rate of heat transfer for the cylindrical analog of the situation depicted in the following figure.

2. Summary of the problem
As shown in the sketch, the solid is in the form of a hollow cylinder and the outer and inner surfaces are maintained at temperatures T1 and T2, respectively.
The ends of the cylinder are insulated so that heat can flow only in the radial direction.

3. Write down your assumptions
No heat source
Constant k

4. Consider a thin cylindrical shell element of thickness ∆r in a long cylinder as shown below

5. Energy balance equation

8. Simplifications

10. Simplifications

11. Compact form of the conduction equation in one direction for all coordinates

12. Examples