09FE14

. M914 Dissertation The Stefan-Boltzmann formula:
σ = 5.67E-008 W/m²K4 is the Stefan-Boltzmann Constant q(W) = σ T*T*T*T * A is the Stefan-Boltzmann formula

In interstellar space: Temperature Blackbody emission to -273C space

T(K): T(C): T*T*T*T W/m2

surface infraemissivity 273 0 5554571841 315 Real surface emission to -273C space

(Solar absorption as=1) 283 10 6414247921 364 W/m2 Real surface emission to +33C surrounding

e: 290 17 7072810000 401 W/m2 Convective and conductive heat-transport

1 306 33 8767700496 497 497 0 W/m2

1 365 92 17748900625 1006 1006 509 0 blackbody

.9 375 102 19775390625 1121 1009 562 0 wood, paper, glass, masonry, nonmetallic paints

.5 434 161 35477982736 2012 1006 757 0 Aluminium paint

.2 545 272 88223850625 5003 1001 901 0 Aluminium-coated paper, polished

.1 648 375 176319369216 9998 1000 950 0 Aluminium sheet, plated Nickel oxide stainless steel

.05 771 498 353360102481 20037 1002 977 0 Aluminium foil, bright

Some radiation and heat related reflections:

Under sunlight irradiation (1kW/m², peak wavelength 500nm visible), ordinary materials absorb almost all the incoming radiation, reflect only 5-10% .
To re-radiate that absorbed energy to the neighbourhood ordinary materials with emissivity of 0.9 have to raise their temperature to 100C,
painted surfaces with e=0.5 shall reach 160C to re-radiate 1kW, aluminium-coated paper (e=0.2) 272C, aluminium-sheet (e=0.1) 375C,
bright aluminium-foil (having the lowest emissivity e=0.05) will reach almost 500C when radiating 1kW!

All that is assumed to happen in the interstellar space, black coated front-side absorbing 100% of insolation, covered with transparent Polyethylene
(e=0.1) that prevents infrared radiation to escape from the front-side, thus emitting all the absorbed energy only from the backside.

On Earth's surface there is additional radiant heat source (with no effect behind the transparent Polyethylene covert on front-side, only at backside) :
the surrounding objects that radiate 300-500W/m² (peak wavelength 10um infrared) at 0-30C ambient temperature (insolated objects radiate far more).
However, there is convective air-cooling on the front-side of the surface, and conductive heat-transport to the backside of any surface.
So could real surface temperatures reach approximately 60, 70, 80, 90 and 100C respectively, lower than the theoretical ones in interstellar space:

On EARTH's surface: Temperature Blackbody emission to -273C space

K: C: T*T*T*T W/m2

surface infraemissivity 273 0 5554571841 315 Real surface emission to -273C space

(Solar absorption as=1) 283 10 6414247921 364 W/m2 Real surface emission to +30C surrounding

e: 290 17 7072810000 401 W/m2 Convective and conductive heat-transport

1 303 30 8428892481 478 478 0 W/m2

1 328 55 11574317056 656 656 178 822 blackbody

0,9 333 60 12296370321 697 628 197 803 wood, paper, glass, masonry, nonmetallic paints

0,5 343 70 13841287201 785 392 153 847 Aluminium paint

0,2 353 80 15527402881 880 176 81 919 Aluminium-coated paper, polished

0,1 363 90 17363069361 985 98 51 949 Aluminium sheet, plated Nickel oxide stainless steel

0,05 373 100 19356878641 1098 55 31 969 Aluminium foil, bright

Each surface is adjacent to some mass, say 10cm concrete (having 200kg weight, in thermal capacity equivalent to 40 litre water),
hence the daily average 4kWh solar irradiation will raise the temperature of the whole block to 100C (if no heat losses).
However, convective air cooling (100m3 air heated by 10C will bring a quarter of the daily accumulated heat away)
heat conduction and thermal radiation lower the daily accumulated heat of the insolated concrete plate to about 50C.

If the insolated surface of the 10cm thick concrete plate (as=0.9) reaches 50C, the acclimatised backside held at 20C, heat transport will be 600W/m².
The usual 24W/m²K heat transfer coefficient for outer walls with 35C outer temperature results 360W/m² heat dissipation by convection and radiation.

By applying 10cm rock-wool instead of concrete, heat transport will drop to only 12W/m² at the same temperature gradient,
only negligible fraction of the irradiated energy can be transported to the backside, the irradiated surface will overheat from 50C to 75C.
Due to this overheating the temperature gradient will raise to 55C instead of 30C, hence heat transport will increase to 22W/m² in place of 12W/m²,
radiative and convective heat-dissipation will raise from 360W/m² to 960W/m² (applying the usual 24W/m²K outer surface heat transfer coefficient).

Central European building with 300m2 outer surface, 100m2 of it insolated:

But even the very small amount of transmitted heat (22W/m²) will result in 2kW cooling demand for 100m2 insolated surface.
Heating demand for 25C temperature difference during winter is 10W/m² with 10cm rock-wool insulation,
hence for a 300m2 outer wall house it results a 3kW heating demand, or 72kWh daily energy consumption.
Calculating with average 1 hour daily solar insolation during winter the energy consumption will be slightly lowered by 2+3kWh
to 67kWh (although the only one hour of winter sunshine could deliver 100kWh heat to the 100m2 insolated house walls!).
Assuming 100-150 winter days, yearly heating demand will reach 7-10.5MWh (23-35kWh/m² to 300m2 walls),
100-150 summer days with average 10h daily insolation will yield 2-3MWh (20-30kWh/m² to 100m2 insolated walls) yearly cooling demand.
Additionally there is the cooling demand for 5C temperature difference during summer for 300m2 10cm rock-wool insulated walls:
2W/m² result 600W continual cooling demand, or 14.4kWh daily energy consumption, about 1-2MWh (3-6kWh/m² to 300m2) yearly cooling demand.

Radiation Control Coatings
(RCC, high solar reflectance, high thermal emittance)

Applying reflective coating (rs=80% in place of 10%) on 100m2 insolated surface will result in 400W cooling demand in place of 2kW,
hence 100-150 summer days with average 10h daily insolation will yield 400-600kWh (4-6kWh/m² to 100m2 insolated walls) yearly cooling demand.
That reflective coating will in winter lower the insolation gain by 1.6kWh daily, increasing by 0.16-0.24MWh (1.6-2.4kWh/m²) yearly heating demand.

So in place of 3-5MWh cooling demand for 300m2 walls we will get 1.4-2.6MWh just by applying a standard white calcium-carbonate paint.
The winter disadvantage of reflective coating is just 10% of the summer cooling savings, hence it's worth to paint the insolated surfaces white even in Central Europe. Reflective coatings bring far more advantage in the Mediterranean, Tropical and Arid climate zones:
they can halve the 6-10MWh yearly cooling load of the 300m2 outer surface, 100m2 insolated, 10cm rock-wool insulated house.

Wavelength dependent reflectance

Focusing now on reflectivity, one has to take in account that for different coatings reflectivity is wavelength dependent:
Half of the solar energy comes in the visible region (300-700nm), the other in the near-infrared (1-3μm). White colour has a 80% reflectance of solar
spectrum, darker colours have less reflectance, black has only 5% reflectance. In the thermal infrared spectrum (3-40μm) all usual colours are “black”.
Thermal infrared ( far-infrared ) radiation spectrum extends from wavelengths of 3 μm to 40 μm for surface temperatures between +100 �C and –10 �C.

Some substances generate high reflectance in the near-infrared (1-3μm) region: TiO2 white paint reflects solar spectrum much better than the usual
carbonate white, and there are some special additives like InfraCool or EnergyStar which enhance even more the near-infrared (1-3um) reflectivity.
Hollow glass/silica/ceramic microsphere additives of 10-100μm diameter in acrylic colours claim to have high near-infrared reflectance.
Darker colours (coming from an unfavourable starting point) will benefit much more than lighter ones from the use of the near-infrared reflecting paints.
Pure white heat reflective paint will only provide a slight benefit to using a conventional white paint.

There is a 8 - 14 �m transmission window in the atmosphere that is relatively independent of the moisture level such that heat can effectively
be removed from the surface, and even the whole building, by radiative transfer processes – this atmospheric window is of crucial importance:
Without the infrared atmospheric window, the Earth would become much too warm to support life, and possibly so warm that it would lose its water,
as Venus did early in solar system history. Thus, the existence of an atmospheric window is critical to Earth remaining a habitable planet.

The SkyCool case
SkyCool (Wojtysiak patent 2002) is a highly reflective white paint combined with a claimed high emissivity of et=at=0.94 at thermal infrared (rt=0.06).
Selective emissivity in the 8 to 13 μm region is achieved, at least in part, with hollow microspheres, but Wojtysiak patent does not describe
the mechanisms for enhancing emissivity. The thermal emissivity of SkyCool coating is just slightly higher than that of usual surfaces having
emissivity of 0.9, hence practically no any advantage from its implementation: all materials use this cosmic window to give heat freely away
without regard of the selective emissivity in the 8 to 13 μm region: even surfaces with uniformly emissivity et=0.9 are forced to give heat in the
8 - 14 �m transmission window away, it is the only leak where heat can escape.

But, if the emissivity of SkyCool is really selective in the 8 - 14 �m transmission window, so this coating should very poorly absorb thermal infrared
( far-infrared ) radiation outside this 8 - 14 �m transmission window, and without hindrance re-emit most of the absorbed energy into the 8 - 14 �m
transmission window (ambient temperature spectral energy peek). Assuming half of the ambient temperature thermal spectrum energy being outside
the 8 - 14 �m transmission window, SkyCool may absorb only 300W/m² instead of 500W/m² thermal radiation of the hot summer-day surroundings
(average thermal absorptivity being at=0.6, seemingly contradicting Kirchoff's law et=0.94, but at=0.6 is meant for broader range 3 - 40�m, et=0.94
is meant in the 8 - 14 �m transmission window).

There is no spectral emittance graph published for SkyCoal, in opposition to the 3M paints being thoroughly analysed in the
„Solar heat reflective paint & coatings using 3MTM Glass Bubbles” publication: The solar reflectance is calculated using the ASTM method E903-96
and reference table for solar irradiance from ASTM G173-03. Thermal emittance is measured according to NF EN 12898 standard using an ABB
B3MEM MB-154S FTIR spectrometer and a gold coated integrated sphere available from SphereOptics.
From the thermal emittance graph one can see, that there is a slight decrease in spectral emissivity outside the 8 - 14 �m transmission window,
hence any coating, even the most usual calcium carbonate CaC33 paint can claim enhanced emissivity e=0.94 in the 8 - 14 �m transmission window.

Application of the SkyCoal reflective coating (rs=80% in place of 10%) on 100m2 insolated surface will result in 400W cooling demand in place of 2kW,
and additional 200W/m² cooling gain if SkyCoal has really selective emissivity in the 8 to 13 μm atmospheric transmission window,
hence 100-150 summer days with average 10h daily insolation will halve the 400-600kWh (4-6kWh/m²) yearly cooling demand due to reflectivity only.
Such a high, doubled cooling gain is not registered for SkyCoal in comparison with other white coatings, so the claim of selective emissivity in the 8 to
13 μm atmospheric transmission window does not hold. Neither a humble 10 or 20% performance gain is registered in comparison with other products.
Cooling gain due to hollow ceramic or glass spheres is only few percent when compared with the calcium carbonate CaC33 paint.

Reflectance of these white coatings starts at the very high 97% in the solar range by steadily decreasing to 50%, then further even to 5% in the
thermal range, quite opposite to the metallic surfaces. At 3-4 �m band there is a temporary increase of reflectance from 5 to 30% and in the 6-6.5�m
band from 4 to 12% for all coatings including the calcium carbonate CaC33 paint too. These two bands of temporary increase of reflectance for
longer wavelengths might be the clue for warmer, condensation and mold-free surfaces, but there is no need for special substances: it will function
equally well with the usual calcium carbonate CaC33 paint too.

And lastly there is very often the claim: „heat-reflective paint with high reflectance in the infra-red to minimise the surface temperature increase”.
Infra-red is here meant to be the solar near-infrared, for thermal infrared all these surfaces are „black”, highly absorptive, no thermal reflectance.

Interior Radiation Control Coating Systems
(IRCCS, far-infrared emittance of 0.25 or less: high thermal reflectance, low thermal emittance)

A low-emittance paint or coating is available for use on interior surfaces to reduce heat transfer between the interior air and the coated surface.
Radiant barriers ( shiny metallic surface with a far-infrared emittance of 0.1 or less) are always installed with the reflective (low emittance)
surface facing an open air space.

Because of a radiant barrier’s low emittance and high reflectance, it can block about 90 – 97% of the radiant heat that strikes the surface,
significantly reducing the total heat transfer in and out of a building.

Radiant Barriers and Reflective insulation products excel in hot climates and often are the first choice for insulation in those regions.
However, they also provide significant benefits in cold climates where they are used alone or in combination with other insulation materials.

Thus, except in sunlight,the colour of clothing makes little difference as regards warmth;
likewise, paint colour of houses makes little difference to warmth except when the painted part is sunlit.
The main exception to this is shiny metal surfaces, which have high reflectance and low emissivities both in the visible wavelengths and in the far infrared.
Such surfaces can be used to reduce heat transfer in both directions; an example of this is the multi-layer insulation used to insulate spacecraft.

09FE14 Dissertation

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