Net Radiation b/w Two Spheres: 1069.3 Watts

[philrainey]

I have a much better shape that is much easier to calculate.Try calculating this one the net radiation between two spheres one within the other a perfect vacuum inbetween.outer Big sphere is 0.13m diameter at 1500 kelvin. the little sphere is .01m diameter at 1550 kelvin. what is the net radiation and in which direction.

so the little sphere area is0.000314metres square at 1550 kelvin=(56.7*10^-9)*(1550^4)*0.000314=102.7 watts.
Total radiation from the big sphere area of0.053066 metres square=(56.7*10^-9)*(1500^4)*.053066=
15232 watts radiating from all the surface of the colder big sphere
but only 7.7% of this radiation hits the little sphere as the rest misses it and hits the big sphere again on opposite side.15232*.077=1172watts so the net radiation is 1172-102.7=1069.3 watts from the colder big sphere to the smaller hotter sphere . I'm led to believe the intenisity of radiation is proportional to the cosine of the angle to the normal to the surface it is radiating from.

 

[philrainey]

by the way they are both perfect black bodies

 

[philrainey]

my calculation is wrong can someone please look at my calculation and logic to show me where I am wrong?

big sphere 0.13m diameter
little sphere 0.01m diameter
taking a point on the big sphere and working out the % of total radiation from that point that hits the little sphere.First the angle from the normal to edge of little sphere
tan(angle)=(0.01/2)/(0.13/2)
=0.07677189
make x the angle from the surface that just hits the inner sphere
x=1.57-0.07677189
x=1.4932281
radiation intensity proportional to cosine to the normal of the radiation or the sine of x
antidifferential of sinx is -cosx
enter angle x=-cos1.4932381
=- 0.0774905
At the normal 90 degrees=-cos1.5706
=0
Area under sin wave between normal and =0--0.0774905
Area under angle that hits small sphere =0.0774905*2
=0.1549869
% of total radiation that hits little sphere=0.1549869/2
=7.749345%
2 been the area under a sin wave for 180 degrees?

big sphere temp=1500 Kelvin
little sphere temp=1550 Kelvin
area of little sphere=0.000314 metres square
area of big sphere =0.053066 metres square

heat radiated by little sphere all that hits big sphere=(56.7*10^-9)*(1550^4)*0.000314
=102.76 Watts
total heat radiated by surface of big sphere=(56.7*10^-9)*(1500^4)*0.053066
=15232.26364 Watts

radiated heat from big sphere to little sphere=15232*0.07749345
=1180.3 Watts
net radiation =1180.3-102.76
=1077.6 Watts?It is wrong

 

[russ_watters]

So you're asking the same question you did two days ago and hoping for a different answer this time? You already acknowledged you don't know how to calculate the effect of geometry, so you know the 7.7% is wrong and I already pointed out that you are misusing the Stefan-Boltzman Law and gave you the correct form of the equation. If you want to know how to derive the law mathematically (as opposed to guessing about how it works), just ask that question - maybe someone can help with that.

 

[philrainey]

O.K I'll do that

 

[philrainey]

It was the shape of the sigsawed surface radiating that was just way too hard for me to calculate the effect of geometry, the sphere in a sphere is much much simplier and I want to find out where the calculation is wrong . The Stefan-Boltzman Law is P=area * emissivity*absolute temperture^4 I don't know how the formula in the website http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan.html is produced or what it is based on, I'm sure it doe not consider the surroundings a perfect sphere. I'm asking for help as to where the logic I used in the calculation is wrong or where the number crunching is wrong.