If we assume an array of 1m x 1m mirrors reflecting sunlight in an Alabama field to a common point, what would be the maximum possible temperature that could be reached?

How many mirrors in this array could be added that would still have an influence on the temperature of the point of focus? Is the temperature increase proportional to number, quadratic?

The mirrors could be angled and positioned far enough away to be protected from the heat.

What are practical implications in an Alabama field

Engineering questions aside, what hard limits might there be in the physics of doing this on earth (other than the second law limiting it to the temperature of the sun which has been discussed). For example, is there a limit imposed by losses through the atmosphere? If we assume constant atmospheric absorption, what does that do? What the atmosphere changes frequency profile?

Related: Also Highest temperature possible to achieve using magnifying glass and sunlight which links to Is it possible to focus the radiation from a black body to make something hotter than that black body?

  • $\begingroup$ Also physics.stackexchange.com/q/184437/123208 which links to physics.stackexchange.com/q/140949/123208 Also see en.wikipedia.org/wiki/Etendue although that article isn't an easy read for the non-specialist. And of course what-if.xkcd.com/145 physics.stackexchange.com/q/370446/123208 $\endgroup$
    – PM 2Ring
    Aug 23 '21 at 7:05
  • $\begingroup$ @PM2Ring ok got it figured out finally. Thanks. I dont take it to be a repeat as the bama reference (to me) had me thinking “what are some practical limits”. Plus asking about number of mirrors. $\endgroup$
    – Al Brown
    Aug 23 '21 at 7:13
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    $\begingroup$ @AlBrown It's not an exact dupe, but it's pretty close. OTOH, I won't mind if this question doesn't get closed. Asking about the physics that limits the temperature, and the general arrangement of the mirrors is on-topic, but asking for details like the number of mirrors is getting into engineering territory. $\endgroup$
    – PM 2Ring
    Aug 23 '21 at 7:21
  • $\begingroup$ I added number of mirrors. Check it out. Like this question. Asked for a reopen. About that reopen issue: note that I dont talk about materials, or the set-up more than a sentence “scaffolding” etc. just power balance and differential $\epsilon,\alpha$ around the object, etc. $\endgroup$
    – Al Brown
    Aug 25 '21 at 4:33

First hard Theoretical Limit:

One limit, even in space with no atmosphere, with almost infinite mirrors.. is the Sun’s temperature. When it reaches the temperature of the Sun, it must be emitting as much radiation at the Sun as the Sun is emitting at it. So its temperature cannot increase further. (The other question’s replies say only that, as only that was asked. Not a duplicate. E.g. no enquiry about...

Number of Mirrors

Assuming we’ll have a sufficient temperature for radiation to dominate convection, the number of mirrors needed grows quickly:

If $\alpha$ is its absorptivity, $N$ is the number of mirrors, $w$ is the power in $1m^2$ of direct sunlight, $r$ is the fraction of light perpendicularly incident (or equivalent thereof), then the total power absorbed is: $$P_{in}=\alpha r wN$$

mirrors heating to white-hot

Power plant near Primm, NV where they point a couple hundred thousand huge, curved mirrors at a water tank and make it white hot:


To maximize temperature, we wait for equilibrium, $P_{in}=P_{out}$. Scanning a solar power site, $w=600 \frac{W}{m^2}$ is a guesstimate for an earth summer temperate-zone. Achieve $r=$~$0.5$. For emissivity of $\epsilon$, and the Stefan-Boltzman constant $\sigma$, power balance is $$N\alpha 300=\epsilon \sigma A T^4$$

By the Kirchoff Theorem, at that point (equilibrium), $\alpha = \epsilon$. Yes good emitters are usually good absorbers, and exactly so at equilibrium.

The previous equation (and common sense) says to minimize surface area, but still catch the whole beam from our $1 m^2$ mirrors. OP made a good call with a $1m$-edged cube, $A=6m^2$. $~ \sigma= 5.67\times 10^{-8}$ in SI (mks). Plugging those in: $$N \approx 10^{-9}~ T^4$$

It’s interesting that a poor absorber/emitter reaches the same temperature as a good one, as both directions of the power balance are affected. However, if we worried about getting up to a $T$ where convection is meaningless, then we’d want high $\alpha ,\epsilon$.

Assuming it hasn’t melted, some points on this curve in (deg-C, $N$ mirrors) are:

$(200, 49) ~(500, 350)~ (1000, 2600) ~(2000, 26.5k) ~(3000,114k) ~(4000, 330k)$

It may seem like $330,000m^2$ of flat mirroring is implausible, until we see what’s been done (well over a million $m^2$ of curved, focused mirroring, see image).

The biggest immediate benefit I can see would be concentrating the incoming light on one side and somehow getting higher absorptivity/emissivity there with a composite or varying surface treatment. If we could get $0.8$ there and $0.2$ on the other five sides, then the effective absorptivity would be $0.8$, but the effective (net) emissivity would be between $\frac{0.8+5(0.2)}{6}=0.3$ and $0.8$ - considering the object’s temperature gradients. If it was $0.5$ for example, then $N$ would go down $38$%.

Any practical limits?

Absent unmentioned constraints, it depends almost entirely on how exacting the mirror specs can be with respect to their shape and positioning (an engineering question so not considered; curving the mirrors would help a lot). Another salient factor is the object’s altitude. For a ground location, there would be less total heat at equal precision.

Adding more sunlight will always add more heat, but the net heat levels off and cannot reach sun temperature through earth’s atmosphere. That’s because once it’s hot enough, the object will be transmitting heat mostly via radiation, at a rate proportional to $T^4$. Even in space, it eventually emits back as much as it takes in. Otherwise we would violate the Second Law of Thermodynamics with heat flowing from something cold to something hot in a reversible process.

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    $\begingroup$ Could it exceed the temperature of the sun? $\endgroup$ Aug 23 '21 at 5:38
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    $\begingroup$ Nope. See above $\endgroup$
    – Al Brown
    Aug 23 '21 at 6:57

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