I attach the spreadsheet I built using the original Auto Ignition Equation given in the text. There is discussion of convection losses in the paper.....
You can see it uses similar data.
The ignition temperature of paper is 247 deg C
The solar image in other configurations can be much less than the example, hence the Heat Flux is amplified.
It would be of great interest to see how "real life" experiments correlate to these mathematical models.
I can't really comment on the applicability of the model in the paper by Shen, Fang & Chow, but I suspect that your examples are pushing the limits of their theoretical model. A couple of critical differences between paper and wood occur to me:
. Paper is much thinner than even a small piece of kindling, so it heats up through its thickness very quickly, even when heated from only one face. Conversely, even though paper is a good thermal insulator, there is enough conductivity through such a thin film to ensure the convection / radiation on the unheated face plays a significant role in the overall heat transfer. (E.g. if you put a paper cup filled with water over a naked flame, it won't catch fire; a wooden cup will certainly char on the bottom face, if not burst into flame, in such a test.)
. Radiation and convection from BOTH faces is important for paper, less so for a "solid" piece of wood, for which the far side will stay much cooler than the heated side.
. For radiant heating of a substantial piece of wood, the radiant heat will also heat the air on the heated face, so the convective cooling effect will be limited - you will be drawing in hot air rather than ambient air. For our example (a converging light beam to a very small disc), there is an "infinite" source of ambient-temperature air to replenish the convective cooling mechanism.
. The surface area to mass ratio is MUCH higher for a piece of paper than kindling, let alone a plank of wood. This means that there is a much higher reaction area available to initiate combustion.
In practical terms - if you take a piece of paper, a twig, a dowel, and a plank, and hold them up close to a heat source, the paper ignites MUCH faster than the kindling, dowel or plank.
If I run my thermal analysis using heat input of 156 W/m2 over the 15 mm disc (to simulate the losses in the solar scope components), I find that the "hot spot" reaches an ambient temperature of about 8 degrees warmer than ambient, after about 20 seconds of heating. From then on, there is no net heating, as the radiant heat is in thermal equilibrium with the convective and radiative losses.
What are your thoughts on the concept of a "Critical Heat Flux" as a possible threshold "rule of thumb"??
With a OTA containing no filters etc the Heat Flux at the solar image can get up to 100KW/m^2 (150mm f10) and even a common ED80 (80mm f7.5) can give 178 KW/m^2 flux.
These are pretty big numbers.....
Even my magnifying glass (75mm, 130mm fl) gives 3,300 KW/m^2
There are so many variables, some of which are very sensitive, and / or can vary over a wide range of plausible values, so I would be cautious about using the following other than as a very indicative guide.
However, for what it's worth, if I assume a "worst-case" convection coefficient of 5 W/m2.K on both sides of the paper (representing still air), my thermal analysis suggests the 15 mm hot-spot will reach an equilibrium temperature of 550K (277 Celsius) with an input heat flux of 13,300 W/m2, and about 600K (327 Celsius) with a heat flux of about 16,600 W/m2.
In practical terms - less than about 10,000 W/m2 (10 kW/m2) applied to a small area of paper SHOULD be low enough to avoid ignition; anything over about 12 kW/m2 would run the risk of ignition.
Take this all with a generous handful of salt - it will be interesting to see how these predictions stack up against some actual testing!
LOL Ken... In Sydney it will be weeks if not months before we see air clear enough to try this properly.
I'd say most of us in Sydney are glued to the media waiting to see if the heart of the Blue Mountains will be burned - there are major fires in progress in the Megalong, Burragorang and Grose valleys and its only a matter of time before these reach the Jamison Valley and the townships along the highway from Mt Victoria right down to Glenbrook.
Right now the wind is ENE putting Mt Victoria & Blackheath in the immediate path this afternoon; the wind is expected to swing west which means the rest of the towns to the east will be at risk.
I feel for you guys up there in NSW/ Q’land....
Stay safe.
It’s 46 deg down here and PowerCor have pulled the plug...no power...promise (!!) it should be on sometime after six pm.
The surface temperature of the sun is approx. 5700 deg C, the diameter is approx. 1.4 million km, the earth sun distance is approx. 150 million km. The highest temperature that you can achieve on earth by any known optical means is something a bit less than 5700 deg. C. the size of the image that you can focus on earth is the diameter of the sun divided the ratio the two focal distances at play here. The, a bit lessedness than 5700 deg. C is the efficiency of the lens over the f ratio of the lens. (I think?)
1.4*10^6/150*10^6= 9.4 mm
for 1m focal length @ f10 the temperature would be approx. 80 percent of 570 degrees
0.8*5700/10=456deg. C.
For .1m focal length @f1 the spot will be .94mm at a temp. of 4,569 deg. C.
For 1m @f1, the spot = 9.4mm at 4,569 deg. C
I think I may have stuffed up somewhere on the role of the f factor, but you can see my line of reasoning for at least the spot size and temperature part of the problem.
Merlin66,
True, I left out the attenuation through the atmosphere. I considered It trivial In the context of the discussion, a dB or so, also the glass lens would attenuate the infra red, which Is about half the total energy. That avoids having to mention the CO2 in the atmosphere, boy, that could cause trouble.
If this not what you meant, give me the correct formula please.
Barry,
Look at messages #21, #11 and #1 below. The formula is given, and the attached spreadsheet gives the input data.
The discussions so far indicate that this is not a perfect solution...hence the practical experiment.
Barry all that remains is to do the experiment - and I definitely will when we have a clear day - but for the past few weeks we’re lucky to even see the sun dimly through the smoke.
I think it is incorrect to assume you can "focus" the temperature of a remote object through an optical system. The temperature of an object is a measure of the energy of the atoms, molecules and sub-atomic particles that make up that object; what we perceive as radiant energy is carried only by photons emitted by the object, not by the excited atoms and molecules themselves. The frequency / wavelength / energy of the emitted photons is a function of the temperature of the object (among other things), but the photons do not have "temperature" in the same way that the atmosphere of a star does.
What we can do is measure the Sun's radiant power (W/m2) and focus this into a smaller area to increase the energy flux. We can then try to work out what that focussed energy does - by way of heating the target and surrounding air, and being re-emitted by reflection, radiation, and convection. That is what merlin66's cited paper and spreadsheet attempt, and my thermal analysis tries to better allow for the thermal properties of paper vs wood (on which the cited paper is based).
