Finned tube heat exchanger

The heat transfer area for the fin can be calculated from any standard Heat And Mass Transfer book.

Anyways,

In general the fins have a greater cross section at the base where the tempreature is maximum to allow quick heat transfer and the area gradually reduces along the fin length.

Also increasing the fin length unnecessarily wont help because eventually the heat conduction gradually reduces along the fin length. So try to keep your fin thicker at the base and reduce its area of c/s along the length and keep the fin length optimum to avoid wastage of material.

Again the spacing between the two cnsecutive fins must be optimised so as to allow maximum heat transfer rate from the fin to the ambient fluid where heat is rejected.

This will depend on the maximum tempreature at base, the cooling desired and the coefficient of convective heat transfer of the medium where heat is rejected.

The thermal gradient for different type of fins is given in all standard books and you can refer any one of these to cater your needs. The governing equation can be easily found from there which fits your operating conditions.

All the Best.

Yeah well said Rohan