Heat transfer through fins on a cylinder

[gfd43tg]

Homework Statement

A copper tube has a 2 cm inside diameter and a wall thickness of 1.5 mm. Over the tube is an aluminum sleeve of 1.5 mm thickness having 100 pin fins per centimeter length. The pin fins are 1.5 mm in diameter and are 4 cm long. The fluid inside the tube is at 100 ºC, and the inside heat transfer coefficient is 5000 W/m2 K. The fluid outside the tube is at 250 ºC, and the heat transfer coefficient on the outer surface is 7 W/m2 K. Calculate the heat transfer per meter length of tube. Take k = 204 W/m K for the aluminum and k =386 W/mK for the copper tube. You may ignore the heat loss from the fin tips.

Homework Equations

The Attempt at a Solution

This problem is pretty tough, but basically what I did was this. I tried to find the surface area of all the fins bases per unit length, and then subtract that from the total surface area per unit length. Then I said the convective heat resistance from the outside is the outside heat transfer coefficient times the exposed area plus the area of the fins that protrude from the cylinder. I added all of them in series with the multiple layers and inside convection to find total resistance and divided overall temperature difference by the total resistance. I am just unsure if I did the part about finding the exposed area correctly.

 

[Chestermiller]

What fraction of the outside area did you determine that the fins occupy?

The first step is to focus on a single fin. If T is the temperature at the base of the fin, do you know how to determine the rate of heat loss from the fin as a function of T and the outside temperature?

Chet