Derivation process? (Heatsink Fin Heat Conduction Equations)

In summary, the conversation revolves around the derivation of the right side of the last equation and the inclusion of C_2 for the variation of temperature along the fin. The discussion also involves understanding the equivalence between the rate of heat transfer on the tip and at x=0 under steady state, and the origin of sqrt( hpkA)*(Tb-T_inf). The solution provided suggests that the second equation should be adjusted to θb = θ(0) = C1 + C2, and explains the reason for differentiating θ(x) and treating T_inf as a constant.
  • #1
EastWindBreaks
128
3

Homework Statement


upload_2018-4-23_4-36-37.png


upload_2018-4-23_4-37-2.png

I don't understand the derivation of the right side of the last equation.

Homework Equations

The Attempt at a Solution



upload_2018-4-23_5-8-11.png

I got to this point, I also don't understand why it did not include C_2 for the variation of temp. along the fin.
I am guessing the right side is the rate of heat transfer on the tip? which equals to the rate of heat transfer at x=0 under steady state. but i failed to see how they are equivalent, I don't understand where did sqrt( hpkA)*(Tb-T_inf) came from. any help would be appreciated!

 

Attachments

  • upload_2018-4-23_4-36-37.png
    upload_2018-4-23_4-36-37.png
    49.6 KB · Views: 5,730
  • upload_2018-4-23_4-37-2.png
    upload_2018-4-23_4-37-2.png
    48.2 KB · Views: 2,074
  • upload_2018-4-23_5-8-11.png
    upload_2018-4-23_5-8-11.png
    12.1 KB · Views: 1,363
Physics news on Phys.org
  • #2
Your second equation should be θb = θ(0) = C1 + C2, instead of 2. and since C1 = 0, C2 = θb. Now when you differentiate θ(x) wrt x and multiply both sides by -kA at x=0, it should work out OK.
 
  • Like
Likes EastWindBreaks and berkeman
  • #3
Dr Dr news said:
Your second equation should be θb = θ(0) = C1 + C2, instead of 2. and since C1 = 0, C2 = θb. Now when you differentiate θ(x) wrt x and multiply both sides by -kA at x=0, it should work out OK.
thank you, I got it now. the reason we differentiate θ(x) is because T_inf is treated as a constant right? therefore it is same as dT/dx.
 
Last edited:
  • #4
That is correct.
 
  • Like
Likes EastWindBreaks

1. What is the derivation process for heatsink fin heat conduction equations?

The derivation process for heatsink fin heat conduction equations involves applying the principles of heat transfer and using mathematical equations to model the transfer of heat through the fins of a heatsink.

2. Why is the derivation process important for heatsink fin heat conduction equations?

The derivation process is important because it allows us to understand the underlying physical principles behind heat transfer in heatsink fins, and to develop accurate equations that can be used for design and analysis purposes.

3. What factors are considered in the derivation of heatsink fin heat conduction equations?

The derivation of heatsink fin heat conduction equations takes into account factors such as material properties, geometry of the fins, boundary conditions, and heat generation within the heatsink.

4. How do the resulting equations from the derivation process help in practical applications?

The equations derived from the process can be used to calculate important parameters such as temperature distribution, heat flux, and thermal resistance in heatsink fins. This information is essential for designing efficient cooling systems and predicting the performance of heatsinks.

5. Are there any limitations to the derivation process for heatsink fin heat conduction equations?

One limitation of the derivation process is that it assumes ideal conditions and does not account for factors such as heat loss due to convection or radiation, which can affect the accuracy of the equations in real-world scenarios. Additionally, the equations may not be applicable to complex heatsink designs or non-uniform heat sources.

Similar threads

Heat Transfer: Effect of a Heat Fin
    • Introductory Physics Homework Help
Replies
4
Views
1K
Heat transfer between objects by conduction
    • Engineering and Comp Sci Homework Help
Replies
13
Views
1K
Steady state heat flow: radiation and conduction
    • Introductory Physics Homework Help
Replies
5
Views
2K
Rate of conduction heat transfer with different hot-side and cold-side areas
    • Thermodynamics
Replies
8
Views
1K
Calculate the efficiency of a furnace’s heat transfer process
    • Introductory Physics Homework Help
Replies
9
Views
2K
I Change of variables for this derivative in a heat transfer equation
    • Calculus
Replies
2
Views
1K
Heat conduction and phase changes
    • Introductory Physics Homework Help
Replies
2
Views
1K
Which Container is Hotter After 30s? Exploring Heat Transfer
    • Introductory Physics Homework Help
Replies
8
Views
2K
Heat transfer to solution per litre
    • Introductory Physics Homework Help
Replies
2
Views
725
Conservation of energy (Waterfall)
    • Introductory Physics Homework Help
Replies
11
Views
2K

Hot Threads

Recent Insights

Back
Top