What are the limits....

[KrazyHorse]

You're also ignoring the fact that you only heat one side of the ball. You should use as the are of the ball pi*r*r instead of 4*pi*r*r when you figure out your maximal effective mirror size. You still have to use 4*pi*r*r when you're talking radiative cooling, however.

Let a be the absorption coefficient
e be the emission coefficient
m be the mass of iron you wish to heat in kg
p be the mass density of iron in kg/m^3
s be the power per unit area of solar radiation at the earth's surface in W/m^2
c be the heat capacity of iron in J/(K*kg)
sigma be the stefan boltzmann constant
n be the number of maximally sized mirrors
and T0 be the ambient temperature (i.e. starting temp of metal) then a good approximation is:

dT/dt = [pi*(r^2)*n*s*a - 4*pi*r^2*sigma*(T^4)*e]/(m*c)

with r = (3m/4*pi)^(1/3)

unfortunately I have no idea how to integrate 1/(1-x^4) to get an analytic solution...

[aneeshm]

Originally posted by KrazyHorse

You're also ignoring the fact that you only heat one side of the ball. You should use as the are of the ball pi*r*r instead of 4*pi*r*r when you figure out your maximal effective mirror size. You still have to use 4*pi*r*r when you're talking radiative cooling, however.

I didn't ignore that, that''s what I've done, too. I've taken the area to he heated to be pi*r**2, and I've taken the area for heat loss to be four times that.

[Lul Thyme]

Originally posted by KrazyHorse

with r = (3m/4*pi)^(1/3)

unfortunately I have no idea how to integrate 1/(1-x^4) to get an analytic solution...

Partial Fractions?
1/(1-x^4)=A/(1+x^2) + B/(1-x) + C/(1+x)

where A=1/2 and B=C=1/4 unless I made a mistake in the calculation...

[KrazyHorse]

I hate partial fractions.

[KrazyHorse]

I assumed it was something like that, but for some reason I thought it wouldn't work out because of the A term.

I guess you can do integration by parts on that. Or a clever trig substitution.

[Lul Thyme]

Originally posted by KrazyHorse
I assumed it was something like that, but for some reason I thought it wouldn't work out because of the A term.

I guess you can do integration by parts on that. Or a clever trig substitution.

yeah in the reals the 1/1+x^2 is some trig function or something.
It's probably in a basic table.

Btw I was going to edit my previous post to add a snide comment about this being college level math and questioning physicist's ability at integrating basic function, but my browser was being slow and by the time I was able to load it, it was too late

I like partial fractions because in a way they have nothing to do with integration, and you can use them for many things.

[chequita guevara]

There was a myth busters episode about this. They were trying to recreate Archimedes mythological death ray. No one could do it.

[aneeshm]

Chegitz - this is not an attempt to re-create Archimedes' death ray. This is a proven technology that is known to work.

[KrazyHorse]

Originally posted by Lul Thyme


yeah in the reals the 1/1+x^2 is some trig function or something.
It's probably in a basic table.

I think it's an arctan or smething. Not sure.

[KrazyHorse]

I used to be better at integrating, but it's been a while since I've had to integrate anything challenging analytically

[Ramo]

I think it's an arctan or smething. Not sure.

Yep.

[Lul Thyme]

BTW Krazyhorse (or anybody interested), partial fractions "always work", if you know all the tricks.
That is if you have a polynomial that factors (f=g*h), you can always find polynomials m,n such that 1/f=m/g+n/h.

This means that any rational function can be integrated in principle, and in fact in practice if you know the roots.