Heat transfer in a fin with constant volume

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Discussion Overview

The discussion revolves around the heat transfer characteristics of a cylindrical fin with a constant volume as its length varies. Participants explore the implications of changing the fin length on heat transfer efficiency, perimeter, and cross-sectional area, while considering the relationships defined by relevant equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that increasing the length of the fin could lead to various outcomes for heat transfer, including increases, decreases, or a combination of both.
  • Others question how the area and perimeter of the fin change with length under the constraint of constant volume, suggesting that an increase in length results in a decrease in perimeter and cross-sectional area.
  • A participant provides a mathematical relationship for the efficiency of the fin, indicating that efficiency is related to the parameter m, which depends on the perimeter and cross-sectional area.
  • It is noted that with constant volume, the radius of the fin is inversely proportional to the square root of its length, prompting further inquiry into how this relationship affects fin efficiency and heat transfer rates.

Areas of Agreement / Disagreement

Participants express differing views on the effects of increasing fin length on heat transfer, with no consensus reached on whether heat transfer increases, decreases, or follows a more complex pattern. The discussion remains unresolved regarding the specific impacts on efficiency and heat transfer rates.

Contextual Notes

Participants acknowledge that the relationships discussed depend on the assumption of geometric similarity and the constraints of constant volume, which may limit the applicability of their conclusions.

Who May Find This Useful

This discussion may be useful for individuals interested in heat transfer principles, particularly in the context of fin design and efficiency analysis in engineering applications.

Kevin Spears
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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.
 
Last edited:
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Is this a homework problem?
 
Chestermiller said:
Is this a homework problem?
No its not.
 
For constant volume how does the area change with fin length, and (assuming geometric similarity) how does the perimeter change?
 
Chestermiller said:
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)
 
Last edited:
Kevin Spears said:
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?
 
Chestermiller said:
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.
 
Last edited:
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?
 

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