The amount of radiant energy Q passing from the nth to the (n+1)th foil is:
Q = SBconst x A x (e/2-e) x [T(n)*4 - T(n+1)*4]
Between all pairs of foils, Q is the same; Paul in Montreal says: "Each foil will be in equilibrium if the amount of energy it radiates is exactly equal to that which it receives - from both sides".
All foils have the same e, so for all pairs of foils (e/2-e) is the same; Martian's error is to say "e gets smaller".
Therefore for all pairs of foils [T(n)*4 - T(n+1)*4] is the same; Martian's error is to saythis term "must increase proportionately" as "e gets smaller" (e doesn't).
The only thing that differs, for each pair of foils, is a fresh pair of values of Tn and T(n+1) - there are numerous value pairs that make [T(n)*4 - T(n+1)*4] come out the same. Here are a few samples:
See the final column; despite the proper application of fourth-power, [T(n+1) - Tn] comes out at 3K +/- 10% for all foil pairs. That *is* "near enough" linear, as a useful approximation within the limited range 272K to 293K.
Still, never mind that, the point is that the foils' steady-state temps are "near enough" linearly (evenly) spaced,
*not* as Biff, Martian et al have been saying, all bunched up at the hot side, diminishing "near exponentially" (funcrusher) to insignificant for foils 2-6.
Note that this even spacing is independent of e. The same steady-state temp spacing arises, whether the foils are matt black or shiny silver (assuming all are the same). e only tells us how much nett radiant energy passes *as a result of* that sequence of 3K temp differential.
All foil pairs having substantial temp differences between them, *all* foils play a role. If there's still a question "what do foils 2-6 do?", it cannot be on account of a supposedly negligible temp gradient between them, which is Biff's error.
Or is Martian's formula incorrect, or inapplicable to series of foils?
Martian
posted on 01-02-07
Tom,
I think maybe there was a mistake in my interpretation, but not any of the ones you think.
Firstly I never claimed that the emissivity (e) changes for any given material. I commented about the different temperature distributions you could expect between different emissivities varying from e = 1, the "black body" scenario, to materials with very low e values such as polished aluminium.
Where I may have been in error is in thinking that the temperature distribution may be linear even in the case of e=1 foils. On further reflection (ho, ho) I an inclined to believe that the distribution will be non-linear even in this situation. I will consider this further when I am less busy and will get back to you.
I think that the formula I quoted is probably correct but only for a pair of foils and that some far more serious mathematics will be required to deduce the temperature distribution for a series of them.
Please also remember that this is all true only in a vacuum!! It still does not help your earth-bound variant.
fostertom
posted on 01-02-07
Yes, Martian, I now see what you said, and apologise:
>I never claimed that the emissivity (e) changes for any given material. I commented about the different temperature distributions you could expect between different emissivities<
However.... (always a however!):
<As e gets smaller
e/2-e gets smaller by larger increments
which means that [T(i)*4-T(o)*4] must increase proportionally by the same larger increments for the equation to balance.>
It doesn't mean that [T(i)*4-T(o)*4] must increase; it means that Q must decrease; [T(i)*4-T(o)*4] stays the same. In other words, as e gets smaller, radiant nett heat flow Q decreases accordingly.
The (e/2-e) of different materials describes how much each one radiates/absorbs, when subjected to a standard (constant) radiant temp differential.
fostertom
posted on 01-02-07
Correction:
>....when subjected to a standard (constant) radiant temp differential<
should read:
....when subjected to a standard (constant) [T(i)*4-T(o)*4].
The first statement is still "near enough"; the second is exact.
funcrusher
posted on 01-02-07
Fostertom Disproved:
First, you haven't actually addressed the question what layers 2-6 actually do, which is really a specific case of 'What is the effect of multilayers?'
I'll come back to that after challenging your maths.
Take a simple 4 layer system, idealised in outer space, temperatures t1, t2, t3, t4. As the inter-layer heat flow Q is the same and the layers and constants are identical, I think you will readily agree that
and so eg (t1*2+t2*2) > (t2*2+t3*2) AND (t1+t2) > (t2+t3) etc
AND THEREFORE (t1-t2) cannot equal (t2-t3) cannot equal (t3-t4)
IN FACT AMAZINGLY (t1-t2) < (t2-t3) < (t3-t4) must be true if the mega equation is true.
In other words, as you proceed through the layers from the hot side, not only does the temperature decrease, the temperature fall accelerates!!
If you think about it, this has to be the case because, expressed another way, [t1*4-(t1-T)*4] >[(t1-T)*4 -(t1-2T)*4] etc ; where T is an hypothetical equal temperature interval.
QED
As to what the multilayers do:
Comparing say 5, 6 or n layer MFs, each additional layer has the effect of raising the temperature of the hottest layer and reducing the temperature of the coldest layer.
I think also that your assumptions are misconceived. The temperature of the hottest layer (whether space or earth) differs markedly from ambient if it is receiving significant radiation. Likewise the cold nth layer, which radiates into the ambient surroundings. In fact, if the hottest layer is at ambient temperature, it either a perfect reflector or it receives no radiation. The cold layer at ambient must either be in a nil-radiation situation, or a zero emitter.
As I have said before, the effect of convection etc on earth completely negates the mathematical assumptions.
funcrusher
posted on 01-02-07
(continued)...
The presence of convection reduces the radiant heat flow at each layer, as heat is lost through convection to air. Thus every intermediate layer, and the final layer, will be at a reduced temperatures. Convection losses will be proportional to (tn-ta) where tn is the temp of the nth layer and ta is ambient temp. Hence convection losses are skewed to the hotter layers. I hypothesise that in most circumstances the reduction in temperatures through convection is so great that the temperature gradient decelerates rather than accelerates,ie per our empirical experience of life!
