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...
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...
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