Heat transfer in a fin with constant volume

[Kevin Spears]

Assume we have a cylindrical fin which has the effective length of L and its efficiency is given by the equation: $$η=exp(-0.32mL)$$ where $$m=\sqrt{\frac{hP}{kA}}$$ where P is perimeter and A is the cross sectional area of the fin.

If the volume of the fin remains constant, which of the following statements occur by increasing the length of fin?

1. Heat transfer of the fin increases.
2. Heat transfer of the fin decreases.
3. Heat transfer of the fin increases then decreases.
4. Heat transfer of the fin decreases then increases.
5. Heat transfer of the fin remains constant because the volume is constant.

 

[Chestermiller]

Is this a homework problem?

 

[Kevin Spears]

Is this a homework problem?

No its not.

 

[Chestermiller]

For constant volume how does the area change with fin length, and (assuming geometric similarity) how does the perimeter change?

 

[Kevin Spears]

For constant volume how does the area change with fin length, and (assuming geometric similarity) how does the perimeter change?

As the fin has circular cross section (cylindrical fin), and the volume is constant, increasing length causes decreasing in perimeter and cross sectional area.

The relation can be determined by the following equation:

(Cylinder Volume = Length * Circular Cross Sectional Area)

 

[Chestermiller]

As the fin has circular cross section (cylindrical fin), and the volume is constant, increasing length causes decreasing in perimeter and cross sectional area.

The relation can be determined by the following equation:

(Cylinder Volume = Length * Circular Cross Sectional Area)

So, algebraically, how does that affect the fin efficiency?

 

[Kevin Spears]

So, algebraically, how does that affect the fin efficiency?

As the fin efficiency is related to m,and $$m=\sqrt{\frac{hP}{kA}}$$

so we have:
$$m=\sqrt{\frac{2hπr}{kπr^2}}$$

so
$$m=\sqrt{\frac{2h}{kr}}$$

and finally the efficiency is related to m:
$$η=exp(-0.32\sqrt{\frac{2h}{kr}}L)$$

Please note:
1. L is the effective length
2. Volume of Fin is constant (we have a specific volume of a material and we're going to create a cylindrical fin with that) so, L and r are related to each other.

 

[Chestermiller]

At constant volume, r is inversely proportional to the square root of L. So, if L increases by a factor f, how does that affect $L/\sqrt{r}$, and how does that affect the fin efficiency? In terms of the temperature difference, the cross sectional area of the fin, and the fin efficiency, what is the rate of heat transfer?