What is wronge with this picture?

[Longstreet]

So, the thermal radiation power off a surface is $$P=\sigma AT^4$$.

Here I have two cylinders of equal height but different radius. They are stacked on top of each other inside of two cones stacked on top of each other. The idea is that the thermal radiation off of one cyclinder is reflected into the other one.

With this information I know the power off the two cylinders is:

$$P_1 = \sigma(2{\pi}r_1h/2)T_1^4$$
and
$$P_2 = \sigma(2{\pi}r_2h/2)T_2^4$$

Where P is the power, h is the height of the stack, r is the radius of each cylinder, and T is their temperature.

So say I want $$P_2<P_1$$. All I need to do is make $$r_2$$ small enough. Solving I get the simple relation $$r_2<\frac{T_1^4}{T_2^4}r_1$$.

I know this is extreamly idealized, but what is causing $$P_2 \geq P_1$$.

 

[Longstreet]

This is not a homework question. It is a thought experiment that I had that I couldn't resolve. This obveously violates the second law. I'm not seeing where the flow of energy from hot to cold overwhelms the flow from cold to hot. What twisted professor would assign such a problem anyway. In my example it would probably be better to use a cylindrical ellipse with the pipes at the two foci.

I guess it's my assumption that all the energy from one will hit the other. Since they aren't point objects the randomness of the radiation will cause a significant fraction to miss and come back.

 

[Andrew Mason]

So, the thermal radiation power off a surface is $$P=\sigma AT^4$$.

Here I have two cylinders of equal height but different radius. They are stacked on top of each other inside of two cones stacked on top of each other. The idea is that the thermal radiation off of one cyclinder is reflected into the other one.

With this information I know the power off the two cylinders is:

$$P_1 = \sigma(2{\pi}r_1h/2)T_1^4$$
and
$$P_2 = \sigma(2{\pi}r_2h/2)T_2^4$$

Where P is the power, h is the height of the stack, r is the radius of each cylinder, and T is their temperature.

So say I want $$P_2<P_1$$. All I need to do is make $$r_2$$ small enough. Solving I get the simple relation $$r_2<\frac{T_1^4}{T_2^4}r_1$$.

I know this is extreamly idealized, but what is causing $$P_2 \geq P_1$$.

My earlier post was in error. P is the total radiated power. But P is a function of temperature and area (radius). So if you decrease the area (radius) but do not decrease the power, the temperature must increase. In otherwords, the only way you can have more power going from 2 to 1 is to have T2>T1.

AM

 

[Longstreet]

Well, the congecture is that energy would flow from 1 to 2, which would violate the 2nd law if 2 is hotter than 1. My problem was the radiation coming off of 1 is random at the surface (at r1). So it will not necessarily be focused into 2. Some will miss and come back to 1. My assumption was that because of the elliptical nature, all P1 would go into P2, no matter what the sizes of the pipes.

Now, if there was some way to make the radiation emmit more radially from the surface that argument isn't as strong.

 

[lightgrav]

The cold one needs a larger surface Area for your scheme to work
(even initially, even with perfect equipment ... like correct focusing,
and including a perfect reflector-insulator in between them).
So the cold one's radiation will be not-so-well focused as the hot one's
(which will make the hot one not receive quite as much P as you'd like)
and the extra Area of the cold one will absorb more of the stray radiation
(which means, for A_cold = A_hot (T_h/T_c)^4 , cold warms up).

I'm thinking about spheres at the foci of an internally-reflecting ellipsiod.
If there's NO reflector between them the hot one's DIRECT radiation
will tip the balance ; if the reflector shuts off too much radiation,
each side becomes essentially a black body radiating thru the opening.

 

[Andrew Mason]

The cold one needs a larger surface Area for your scheme to work
(even initially, even with perfect equipment ... like correct focusing,
and including a perfect reflector-insulator in between them).
So the cold one's radiation will be not-so-well focused as the hot one's
(which will make the hot one not receive quite as much P as you'd like)
and the extra Area of the cold one will absorb more of the stray radiation
(which means, for A_cold = A_hot (T_h/T_c)^4 , cold warms up).

I'm thinking about spheres at the foci of an internally-reflecting ellipsiod.
If there's NO reflector between them the hot one's DIRECT radiation
will tip the balance ; if the reflector shuts off too much radiation,
each side becomes essentially a black body radiating thru the opening.

I agree. You can ignore my previous post which missed the point.

The issue here is the ability to concentrate the radiation from the larger cylinder onto the smaller cylinder. It is analogous to trying to use a solar concentrator to achieve a higher temperature than the surface of the sun. It cannot be done. See, for example: http://www.ee.ucla.edu/labs/photon/pubs/ey1990sem2123.pdf

AM