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- TL;DR Summary
- Thought experiment where 2 spheres influence each other's temperatures.
Dear all,
in an encounter of an infamous claim by Gerlich and Tscheuschner that the Greenhouse effect is inconsistent with the 2nd law of thermodynamics I came to a simple thought experiment which I wanted to share with you to check my understanding and brush up my knowledge. The thought experiment I tried to calculate through is as follows.
I have a sphere (1) with radius ##r##, acting like a black body at a temperature of exactly ##T_1 = 500 K##. With Stefan-Boltzmann you can calculate the total power emitted, ##P = A \sigma T^4 = 4 \pi r^2 \sigma T^4## with ##\sigma = 5.670374419 \times 10^{-8} W m^{-2} K^{-4}##. Now we place a second sphere (2) with also radius ##r## (for simplicity) at a distance R from sphere 1, with a temperature of exactly ##T_2 = 600 K##. The question is: what happens with both temperatures, in light of the fact they both absorb a small portion of each others radiation?
Let's put some numbers in. I take ##r = 1 \ m## and ##R = 10 \ m## (we want ##r<<R## to simplify the calculations, so this is just on the edge, but it should work). Drawing the picture, one concludes that the emitted power of sphere 1 is spread over a spherical surface ##4 \pi R^2##, and the surface of sphere 2 which absorbs equals ##\pi r^2##. So the fraction ##c## of power absorbed is given by
$$ c \equiv \frac{\pi r^2}{4 \pi R^2} = \frac{r^2}{4 R^2} = 0.0025$$
The power emitted is given by ##P = A \sigma T^4##. Now some energy considerations: The power emitted after we place the two spheres at a distance ##R## apart in a vacuum becomes altered, which I indicate with an apostrophe. For the new equilibrium states of the emitted powers we then obtain
$$P_1'=P_1+c \cdot P_2' \ \ \ , \ \ \ P_2'=P_2+c \cdot P_1'$$
I.e. the power emitted in the new situation is just the power emitted because of the own inner heatings, plus the fraction of power absorbed from the other sphere, which in equilibrium will be emitted again. Solving this gives
$$P_1'=\frac{P_1 + c \cdot P_2}{1-c^2}$$
which can be plugged in to solve for ##P_2'##.
If I now plug in the numbers, I can calculate ##P_1'## and ##P_2'## and hence ##T_1'## and ##T_2 '##, to find ##T_1'=500.65 K## and ##T_2'=600.18 K##. So the initially warmer sphere is also 'heated' (i.e. it absorbs electromagnetic radiation, raising its temperature) by the colder sphere. The total power emitted changes by
$$P_1'+P_2' = \frac{P_1 +P_2}{1-c} = 1.0025... (P_1 + P_2)$$
My questions:
1) Does this calculation (and approximation) make sense? Are these two new temperatures the new equilibrium states of both spheres?
2) Somehow I'm a bit uneasy with energy conservation. Where did the extra 0.25% of energy come from? Doesn't this cause some "runaway effect"?
I have a feeling something goes wrong. Many thanks. :)
in an encounter of an infamous claim by Gerlich and Tscheuschner that the Greenhouse effect is inconsistent with the 2nd law of thermodynamics I came to a simple thought experiment which I wanted to share with you to check my understanding and brush up my knowledge. The thought experiment I tried to calculate through is as follows.
I have a sphere (1) with radius ##r##, acting like a black body at a temperature of exactly ##T_1 = 500 K##. With Stefan-Boltzmann you can calculate the total power emitted, ##P = A \sigma T^4 = 4 \pi r^2 \sigma T^4## with ##\sigma = 5.670374419 \times 10^{-8} W m^{-2} K^{-4}##. Now we place a second sphere (2) with also radius ##r## (for simplicity) at a distance R from sphere 1, with a temperature of exactly ##T_2 = 600 K##. The question is: what happens with both temperatures, in light of the fact they both absorb a small portion of each others radiation?
Let's put some numbers in. I take ##r = 1 \ m## and ##R = 10 \ m## (we want ##r<<R## to simplify the calculations, so this is just on the edge, but it should work). Drawing the picture, one concludes that the emitted power of sphere 1 is spread over a spherical surface ##4 \pi R^2##, and the surface of sphere 2 which absorbs equals ##\pi r^2##. So the fraction ##c## of power absorbed is given by
$$ c \equiv \frac{\pi r^2}{4 \pi R^2} = \frac{r^2}{4 R^2} = 0.0025$$
The power emitted is given by ##P = A \sigma T^4##. Now some energy considerations: The power emitted after we place the two spheres at a distance ##R## apart in a vacuum becomes altered, which I indicate with an apostrophe. For the new equilibrium states of the emitted powers we then obtain
$$P_1'=P_1+c \cdot P_2' \ \ \ , \ \ \ P_2'=P_2+c \cdot P_1'$$
I.e. the power emitted in the new situation is just the power emitted because of the own inner heatings, plus the fraction of power absorbed from the other sphere, which in equilibrium will be emitted again. Solving this gives
$$P_1'=\frac{P_1 + c \cdot P_2}{1-c^2}$$
which can be plugged in to solve for ##P_2'##.
If I now plug in the numbers, I can calculate ##P_1'## and ##P_2'## and hence ##T_1'## and ##T_2 '##, to find ##T_1'=500.65 K## and ##T_2'=600.18 K##. So the initially warmer sphere is also 'heated' (i.e. it absorbs electromagnetic radiation, raising its temperature) by the colder sphere. The total power emitted changes by
$$P_1'+P_2' = \frac{P_1 +P_2}{1-c} = 1.0025... (P_1 + P_2)$$
My questions:
1) Does this calculation (and approximation) make sense? Are these two new temperatures the new equilibrium states of both spheres?
2) Somehow I'm a bit uneasy with energy conservation. Where did the extra 0.25% of energy come from? Doesn't this cause some "runaway effect"?
I have a feeling something goes wrong. Many thanks. :)
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