# Natural convection and fins heat transfer problem

## Homework Statement

4 rectangular long fins are assembled on a 40x40x6 mm aluminum heat sink with the fins being 30 mm long, t=4 mm in thickness and 40 mm wide. The system generates 3 W and the room temp is maintained at 20 c. What is the temperature of the substrate surface that holds the fins, assuming the system is at steady state? The kinematic viscosity of air v=1.5x10^-5 m2/s, specific heat of the gas Cp=1005j/kgk, thermal conductivity of air k=.026w/mk. For the aluminum, k=160w/mk.

hint: you will need to solve this through iterations. Assume a starting temp of substrate surface at 100c.

## Homework Equations

This is part of the problem, i'm not sure which equations I need.

## The Attempt at a Solution

At first i thought i could approach this using Θ/Θb which equals (t-tinfinity)/(tbase-tinifinty) with tinfinity being the room temp and setting it equal to e^(-nx) where n=sqrt(2h/kt) and using this with q=haΔT but with x=0 to get surface temp, this simplfies to 1, which does not help. Next i tried using a u=1/h with q=uaΔT but i have no second equation to itterate against. I/m also not sure whether to use a qgeneration here because the heat is going through an aluminum block, but in this case I have no temperature for either side of the block and therefore nowhere to start, or if because this is at steady state I can assume q=3W.

I'm not asking anyone to go through and give me the exact number, just enough help to determine equations needed and point me in the right direction.
Thanks a lot for your help.

## The Attempt at a Solution

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The solution will be a combination of the heat transfer by the fins and the heat transfer through the alu. heatsink block.

For the fins you can use the "infinite fin" law that you already stated: Θ/Θb = e^-nx
With this you can define the heat transfer from the fins to the air, which is equal to the heat transfer from the block to the fins at the fin base (x=0).

Now you have to calculate the heat transfer through the block using the standard conduction laws.
Assuming there is no temperature gradient in x,y direction only in z (towards the fins) you can state that Q_fins = Q_conduction = -kA dT/dz at z=6 mm.

I hope this helps you to set up your model.