Direct solar absorption and storage in a molten glass thermocline

At high temperature, the effective thermal conductivity of molten glass becomes dominated by its transparency in the UV-to-2-micron passband. In the previous post it was calculated that molten glass at 2000 degrees K, with a 1/e (i.e., -4.3 dB) absorption length of 1 meter in the UV-to-2-micron passband (a loss rate of 4300 dB/km) would have an effective thermal conductivity of about 456 W/m-°K. The correlations for molten glass reported by Laurent Pilon et al., summarized in this chart, indicate an effective thermal conductivity of molten soda-lime glass at 2000 °K of 137 W/m-°K. From that value it can be estimated that the 1/e absorption length of soda-lime glass of commercial purity is about 137/456 m = 0.3 m. That is a figure that comports with everyday experience (e.g., looking through a glass panel edgewise.)

Thermophysical properties of molten glass at high temperature, including effective thermal conductivity, calculated from correlations in Laurent Pilon et al.

Absorption in the UV-to-2-micron passband of glass is almost entirely due to metal ion impurities, mainly iron. Thus improving the purity of ordinary glass by a factor of ten would extend the absorption length to 3 meters and increase effective thermal conductivity to about 1400 W/m-°K—about three times the room-temperature thermal conductivity of silver.

High effective conductivity also suppresses natural convection since natural convection starts out as a wiggle in an isotherm that enlarges by gravitational instability—despite viscosity slowing it down—faster than it can be damped out by thermal diffusion.

Solar radiation is 90% within the UV-to-2-micron passband of glass, so it plays by the same rules as the conductivity-dominating thermal radiation. A net flux of say, 0.4 MW/m2, onto the top of the glass melt will not penetrate any more deeply if it is the result of solar radiation or thermal radiation (aka, effective conduction.) Sunlight has relatively more short wavelength energy, of course, but under the assumption of spectrally uniform absorption in the UV-to-2-micron band, this makes no difference.

Flux is flux. That greatly simplifies modeling.

Solar heating of a glass melt from above stores heat in a stable thermocline. Later radiant cooling of the melt from above "excavates" the stable thermocline removing heat by natural convection from successively deeper and deeper levels until the whole melt is at the "empty" temperature. Temperatures in the diagram are measured relative to empty. Modeled in Energy2D.  Depth 10 m, absorption length 10 m, shown after 6 hours of solar heating.

Effective thermal conductivity of high-purity glass melts

The effective conductivity of high-purity glass melts increases rapidly with temperature because the melt is semi-transparent to its own thermal radiation. At wavelengths shorter than 2 microns a high purity glass melt may permit its own thermal photons to travel some meters, in some cases even tens of meters, before being re-absorbed. At temperatures used for solar thermal storage, 1840 °K to 2070 °K, about a quarter to a third of blackbody radiation is in the below-2-micron passband of glass. Since effective thermal conductivity is proportional to the average distance a blackbody photon travels before it is reabsorbed, the effective conductivity of a glass melt is a function of its level of purity.



At absolute temperature T, blackbody radiation is

σT⁴,

where σ is the Stefan-Boltzmann constant,

σ = 5.7 E−8 W m−2 K−4 .

The derivative of blackbody flux with respect to T is:

 4σT³ W/m2-°K,

 which is the transfer between two blackbody surfaces differing slightly in temperature. At glass-melt storage temperatures only about a quarter of the blackbody radiation is in the below-2-micron passband of the melt, so we are only concerned with a differential flux of about

σT³ W/m2-°K.

If the average temperature is 2000 °K and photons in the passband travel a distance of one meter, the effective thermal conductivity (ignoring actual phonic conduction which will be relatively small) is

(5.7 E−8 W m−2 K−4)(2000 °K)³(1 m) = 456 W/m-°K

Compare the thermal conductivity of silver at room temperature is 429 W/m-°K.

Cooling the thermal mirrors

The thermal mirrors (thermal cap, yellow; thermal wall, orange) are exposed to intense thermal radiation from the furnace opening or oculus (black line.) The black fringes represent 0 suns, 1000 suns, 2000 suns, etc., of back radiation when the the furnace is at the temperature of the sun.

The thermal mirrors—the cap and the wall—are exposed to intense thermal radiation. If the oculus were at the temperature of the sun, the cap would see about 9,000 suns of flux, and the wall about 1,000 suns. Of course, for the sake of efficiency, the furnace will actually operate at a much lower temperature, reducing these fluxes by about a factor of ten. The mirrors will reflect most of this heat, but perhaps about 5% will be absorbed. This absorbed heat needs to be dissipated from these surfaces to keep the mirrors cool—a thermal flux amounting to about 45 suns on the cap and 5 suns on the wall. The thermal wall may use passive cooling, but active convection is needed for the cap. For comparison, a 2-kw stove element heating a 20-cm diameter pot produces a thermal flux of about 64 suns.

Water is too dangerous to use directly above the furnace opening, so this significant cooling must be obtained by convected air. If the cap is segmented into smaller mirrors, each shingled over the other, a chimney extending to the top of the lamp may draft enough air between the mirrors to keep the mirrors cool.