Heat transfer and radiation question help

See https://en.wikipedia.org/wiki/Bisection_method for information on how to use the bisection method if that's what you're stuck with. What are you stuck with, exactly?

Also, please for the love of all that is good send a photograph of the question.

Just the formula to use and how to calculate it.

I have attached the question.

Thanks for the img. The bisection method itself (at least to me) seems iterative: there isn't some formula for the final answer, you just follow it until you're satisfied with the convergence to a solution. I'm sure it'd be great to code with (and I've used a similar method before for root-finding for practice) but I can't see it's use practically.

Anyhow, if you just want to determine the surface temperature of the heat sink, you'll just need to balance out the equation of state for it. In this context, that's satisfying

$P = \dot{Q}_{rad} + \dot{Q}_{conv}$

where your $\dot{Q}_{rad}$ will be given by

$\dot{Q}_{rad}\propto{T^4}$

with $T$ the temperature of the surface of the heat sink. I've introduced dots because I'm talking about a rate of something, i.e. the amount of heat released per unit time from the heat sink, but anyhow, that's a bit pedantic I know...

The exact constant of proportionality depends on if the area $A$ is just one side or both ("effective" area... is that effective over both sides, or just one? The terminology seems a bit ambiguous. I'd say "effective area of the top surface" or whatever else...).

If you assume $A$ is just the total area of the heatsink, then you get something like

$\dot{Q}_{rad} = \epsilon\times{A\sigma{T^4}}$

where all the constants I've used are as typical for this. That's your radiation component. Convection you can get via https://en.wikipedia.org/wiki/Heat_transfer_coefficient . In a steady state, it's just a matter of rearranging it all to solve for $T$... so I'm not sure where this root finding bisection nonsense comes in. Seems like they want you to use it just to see that you can use it, in which case what's the point? It's math, not physics, anyway I digress... Note that for convection, you'll be considering temperature difference.

If you do intend to go through the bisection nonsense they've dished out at you, here's my guide (from what I can understand from the Wiki page on it.)

i) get f. $Q_{tot}$ is just the power of the resistor. The rest you can solve using what I've said above. Just rearrange it so that you have something like $P + etcetc = 0 = f(T_s)$.

ii) ...

iii) Just consider the eq and sketch the graph. You should see where the derivative is positive, negative, etc. Anyhow, sketch the graph.

iv) The graph should have two roots (as they say, apparently) and one of them will be unphysical, i.e. it might lie in the domain $T<0$ or otherwise might just not be realistic for your setup. This one is one you can ignore.

v) Look at the "valid" root and the domain that it occupies. You might know that it lies somewhere above 0, but below 10,000. Use the bisection method and narrow the domain down.

If you know that the value switches from positive to negative (for f) in this range, where does that happen? Is it between 0 and 5000, or 5000 and 10000?

If the former, you look at the former domain. Split it again. 0 to 2500, 2500 to 5000.

Which part flips? First or second? Focus the one that does. Split it. Do this again.

See the wiki page if you're stuck with it, it actually does a good job. You just need to keep going until you find the point where the value of $f(T_s)$ flips, which will let you find the solution you want.

Thanks for the help. Just wondering how could you code it via Matlab please.

You're doing this as a coding assignment? That makes it way easier!

The wiki page has some pseudocode you can base it on, but this also exists https://www.mathworks.com/matlabcentral/fileexchange/28150-bisection-method-root-finding

Thanks for the help. do you know how to do this question please