Derivation process? (Heatsink Fin Heat Conduction Equations)

[EastWindBreaks]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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!

 

[Dr Dr news]

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.

 

[EastWindBreaks]

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.

 

[Dr Dr news]

That is correct.