I’m joining this conversation quite late, but here’s my $0.02 worth:
The temperature that the paper will reach as a function of time is a non-linear relationship between heat-in (from the concentrated sunlight) and heat-out, from conduction, convection, and radiation, together with the thermal properties of the paper (thickness, density, specific heat, and conductivity).
Typical thermal properties of 80 gsm white paper might be around:
• Thickness: 0.1 mm (a 500-sheet ream is about 50 mm thick)
• Density: 800 kg/m3
• Specific Heat: 1400 J/kg.K
• Conductivity: 0.05 W/m.K
Heat-in can be calculated fairly simply by starting with the assumed solar irradiation of 1,000 W/m2, multiplying by the concentration factor of the lens, and then allowing for losses. For the example given, we arrive at something like 100,000 W/m2 over a 15 mm disc, or 17.7 watts total (without losses), as an upper bound.
Conduction will be very small, as paper is a good thermal insulator – but it can be allowed for in the thermal analysis.
Radiation is a function of the temperature of the paper above ambient, and the nature of the surface – we can assume it is a classical black-body (on both faces) as a first approximation. The hotter the paper gets, the more heat it re-radiates (from both faces), reducing the rate of temperature rise, as the net heat gain (W/m2) falls as the temperature rises.
Convection is the biggest unknown, as it is very dependent upon the air temperature, airflow conditions, and the orientation of the paper.
If the paper is held vertically, natural convection cells will allow free convection to take heat away from both faces. If it is horizontal, the convective heat transfer from the bottom face will be very low (the buoyant hot air “bubble” will be held up against the bottom face of the paper), and for the top face, the heat transfer will be less efficient than when in the vertical orientation.
The presence of any air drafts can change the convection coefficient dramatically – typical values can be as follows:
• Free convection – vertical face in still air: 5 W/m2.K
• Forced convection - Low speed of air over a flat surface: 10 W/m2.K
• Moderate air speed over a flat surface: 100 W/m2.K
• Moderate air speed over a curved surface: 200 W/m2.K
That’s a 20-fold increase (or more) of heat transfer from convection to air, depending on the ambient conditions!
Ref:
https://www.engineersedge.com/heat_t...nts__13378.htm
When I run some basic transient heat analysis, assuming 17.7 W over a 15 mm disc, and allowing for conduction, radiation and convection, I get the following indicative times for the paper temperature to reach 600 K (which should be enough to cause ignition):
• Convection Coefficient < 20 W/m2.K: 0.4 seconds
• Convection Coefficient 100 W/m2.K: 0.65 seconds
• Convection Coefficient 125 W/m2.K: 0.9 seconds
• Convection Coefficient 150 W/m2.K: Not applicable – the convective heat transfer is sufficient to keep the temperature below 600 K
If I repeat the analysis, but limit the heat input to say 10 watts total (~ 40% losses), I get the following indicative results:
• Convection Coefficient < 20 W/m2.K: 0.75 seconds
• Convection Coefficient 50 W/m2.K: 1.0 seconds
• Convection Coefficient 100 W/m2.K: Not applicable – the convective heat transfer is sufficient to keep the temperature below 600 K
It seems therefore that a 150 mm f10 OTA is indeed capable of igniting a piece of paper in about 1 second or so – but it can also be prevented from igniting depending on air conditions, losses, etc.
As I said – it’s a very non-linear relationship, and the efficiency of convective heat exchange from paper to air will be a critical factor in determining whether the paper will ignite.
