Is beam-shaping necessary?

A Fresnel Veselago lens, without any beam-shaping, can produce an annular focal zone of modest concentration from a solar highbeams field.

Beam-shaping (reshaping divergence) at the Fresnel Veselago lens can potentially produce very high concentration at the oculus—and thus reduce thermal losses from the hot space—but this would come at a cost of complexity and reflection loss. Is this a marginal improvement we could postpone for later?

Since reflection losses in beam-shaping with lenticular lens arrays will probably subtract at least of 5% of the total power, and improving concentration at the oculus by a factor of four would reduce the radiant heat percentage, q, by the same factor, the improvement offered by beam-shaping is something like

0.75q - 0.05.

Assuming we would put off for now a 10% improvement in system power in the interest of simplification, that is, if:

0.75q - 0.05 < 0.1,

or q < 20% .

Radiant flux in the hot space at the 3/4-full temperature of 1936 °K is 800 suns (calculations included in the figure above too-generously assume radiant transfer at the empty temperature of 1250 °C = 1523 °K) this becomes a 20% loss when the incident flux is 4,000 suns. From the diagram above it looks like that is probably about four times more concentration than can be achieved at an annular focal zone without divergence reshaping. Adding a 2D CPC concentrator to the annulus or 2D radiation traps for thermal radiation escaping at wide angles might effectively double the concentration at the annulus to 2,000 suns, but that would still leave us a factor of two away, and we would really rather be at 8,000 suns so that heat loss is reduced to 10%.

A better approach is to bring light directly to a circular focal zone. Calculations in the diagram suggest we can get above 3500 suns without beam-shaping, and additionally we can  block much of the radiant heat loss by suspending an elliptical mirror over the oculus. (Sunlight is shown directly striking the boiler tubes, but this could be avoided by slanting the tubes, arranging them on the inside of a cone rather than a cylinder.)

A Fresnel Veselago lens, without any beam-shaping, can produce a circular focal zone with high concentration.

Is beam-shaping necessary? Likely not.

Improved model of thermal flux on the glass lake thermal storage unit

Thermal flux values at glass lake thermal store.

This design reduces the thermal flux at the boiler's water wall to 210 suns, the discharging flux at the lake surface is smaller than that because the lake has 1.5 times the area of the water wall. A much wider range of temperatures is used this time, 550 °C (1250 °C - 1800 °C) but the thermal stratification that occurs during charging allows only about 52% utilization of the glass lake's theoretical storage capacity. The depth of the glass lake and its optical absorption length are both 8 m. Two thirds of the peak solar power of 7.5 GWth goes to storage, giving a plant capacity factor of about 0.75.

Thermal flux at the glass lake

Thermal flux on a glass lake solar thermal store.

Assuming an average annual rated power of 51 W_e per m² of mirror area, peak thermal flux will be about 4 x 51 x 0.71 / 0.4  = 357 W_th per m² land area. Concentrating this flux on a circle having diameter equal to the optics height, H, would give a land-area concentration factor of 28 x 28 = 784, producing a thermal flux of 784 x 0.357 suns = 280 suns. To discharge the lake at 250 suns requires a charging flux of 500 suns, and, since one third of the incident flux is diverted to power generation, a total flux on the lake of 750 suns. That implies a lake with a diameter of 0.61H .

To have the same 250-sun flux at the water wall of the boiler and at the surface of the discharging lake requires the two areas to be equal, and thus the height of the water wall must be one-quarter the diameter of the lake (see diagram below.)

For the area of a cylinder's wall to equal the area of one of its end caps, the height of the wall must be one-quarter the cylinder's diameter.

For an approximately 1 GWe target design, with R_f = 2660 m, and H = 190 m, the lake diameter will be 0.61 x 190 m = 116 m, and the height of the boiler's water wall will be 116 m /4 = 29 m. To accommodate 2.5 times the thermal flux that was modeled in the previous post's thermal simulations requires a lake 2.5 times as deep, and, to keep the same maximum temperature, 2.5² = 6.25 times the optical absorption length. However, that would give a 25m-deep lake with a absorption length of 63 m—so something's got to give.

Charging and discharging transparent glass-melt solar thermal storage

Molten glass thermal storage after 6 hours of solar charging from empty = 0.

Molten glass thermal storage after 6 hours of radiant discharging from full = 250.

The two simulations above were run in Energy2D for a 10-meter depth of molten glass with a 10-meter optical absorption length, charging at 200,000 W/m2 and discharging at 100,000 W/m2. Charging produces a stable thermocline in the melt, while discharging drives natural convection that maintains a nearly uniform temperature within the convecting layer. Surprisingly perhaps, despite the high viscosity of molten glass, the natural convection during discharge is fully turbulent. The discharge simulation above had to be run at 1/10 earth gravity to keep convective turbulence at a scale the model could handle.

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.

High-purity glass melts for solar thermal storage: how pure? what cost?

High-temperature, thermal buffer storage increases the capacity factor of a solar electric plant and thus reduces the cost of thermo-mechanical conversion and electricity generation and transmission. As power sources like wind and photovoltaic—which lack access to inexpensive energy storage—become more common in the utility mix, an increasing value of high-temperature thermal storage will be permitting  time-shifting daily output to seek better electricity prices. Another value of buffer storage, which may or may not be minor, is reducing the thermal cycling of the expensive high-temperature conversion equipment. Here, I will only attempt to assess the first value, the value due to increased capacity factor

Dirk Pauschert in "Study of Equipment Prices in the Power Sector" estimates the cost of an 800 MW supercritical coal-fired power plant at $1960/kw (2008 US$). While that includes costs specifically associated with coal combustion, it also does not include the significant transmission costs associated with a remotely sited solar plant—so I will not apply a correction, $1960/kw it is.

I assume adding 20 hours of storage allows a solar plant to improve its capacity factor from 0.25 to 0.75 (without significant increased losses.) Thus a plant that would have needed 1 kw of turbine/generator capacity with no storage, only needs 0.33 kw of turbine/generator capacity with 20 hours of storage (i.e., having a capacity factor of 0.75 instead of 0.25, it has three times as long to generate the same kilowatt-hours.) That saves two-thirds of the original $1960 cost of the plant, a savings of $1313. With a heat-to-electricity conversion efficiency of 0.40, the thermal storage needed achieve that savings is:

 (20 hours) * (0.33 kw_e) / (0.40) = 16.5 kw-hr_th

Given that two other sources of value were ignored, a conservative estimate for the value of thermal storage is:

($1313) / 16.5 kw-hr_th  =  $80 / kw-hr_th.

Quoting an earlier post about thermal storage in high-purity glass melts:

From Pilon et al., the specific heat, c, of molten glass between 1000 °C and 2000 °C is about 1231 J/kgK...

Another previous post found that the practical delta-T for molten glass storage (bounded by a practical maximum temperature for the insulated roof of the furnace and the need for intense radiative transfer to the boiler tubes) is 230 °C. So a 230 °K storage range stores 230 * 1231 = 283,000 J/kg.

Converting that storage density to kw-hr_th/t:

(283,000 J_th/kg) * (1000 kg/t) * (2.8 E-7 kw-hr_th/J_th) = 79 kw-hr_th/t

Thus the economic value of one ton of high-purity glass employed in this way is:

($80 / kw-hr_th) * (79 kw-hr_th/t) = $6,340 / t

This seemingly high value per ton is not out of line with more familiar high-temperature options.

Table comparing cost and performance of high-temperature thermal storage materials. Image quoted from Xiangyu Meng et al., "Theorectical Investigation of solar energy high temperature heat storage technology based on metal hydrides."

For example, the table above quotes nitrate salts as costing $1700/t, and states that they store 77 kw-hr_th / m³, or 41 kw-hr_th/t —about half as much thermal storage per ton—and at lower temperature than molten glass (which signifies lower conversion efficiency.)

To quote Reiner Haus et al. in "Assessment of high purity quartz resources,"

Beneficiation of raw quartz into refined high-purity products involves several refinement steps which need to be adapted to effectively minimise the specific impurities of the individual raw quartz feed to comply with stringent end-use specifications (Haus 2005). As a result, high purity quartz with total impurity levels less than 20 ppm may be achieved so creating a highly valuable raw material which commands up to 5 EUR/kg.

That is, 20 ppm high-purity silica is currently available at $7,000 / t.

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Optical attenuation in glass-melt solar storage is somewhat similar to attenuation in optical fibers, about which much is known. The spectral attenuation graph for silica fiber reproduced below, shows that, barring metallic ion impurities, there are two different sources of attenuation. At short wavelengths the attenuation is mostly due to Rayleigh scattering; at infrared wavelengths, most of the attenuation is due to OH impurities, and eventually, in the longer infrared, attenuation is due to the tail of the absorption Si-O bonds of the silica itself.

Rayleigh scattering and OH absorption in silica optical fibers. Image quoted from Giusy Origlio, "Properties and Radiation Response of Optical Fibers: Role of Dopants." 

In glass-melt  thermal storage we need much of the solar energy to penetrate deep into the melt to stir its lower half. We do not need particular wavelengths reach the depths. The figure below shows that about half of the energy in solar radiation is shorter than 0.75 microns—and thus more than half of solar energy lies in the region where attenuation is mainly due to Rayleigh scattering.

Percentage of solar radiation shorter than a given wavelength. A wavelength distorted scale quoted from www.powerfromthesun.net .

That is good news because Rayleigh scattering, which is due to thermal inhomogeneities in the glass matrix itself, depends on temperature not on purity. Thus the very difficult problem of reducing the "water peaks" caused by OH ions, which demands extreme purities in glass for optical fibers, does not arise in glass melt storage. For solar glass-melt thermal storage, the easier problem of reducing metallic ion impurities, Fe in particular, reigns instead.

Quoting the Wikipedia article on Rayleigh scattering:

Rayleigh scattering is an important component of the scattering of optical signals in optical fibers. Silica fibers are disordered materials, thus their density varies on a microscopic scale. The density fluctuations give rise to energy loss due to the scattered light, with the following coefficient:^[7]

where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant, and β is the isothermal compressibility. T_f is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material. 

Fictive temperatures for silica glass range from 1373 °K to 1773 °K (1100 °C to 1500 °C); by comparison the temperature range in a glass-melt storage is  1840 °K to 2070 °K (1570 °C to 1800 °C). Since the dependence of scattering on absolute temperature is only linear, there should be about a 25% increase in Rayleigh scattering in a thermal storage glass-melt as compared to an optical fiber.

Since we are now utilizing only half the of solar spectrum to heat the bottom of the melt, short-wavelength attenuation in the melt can only be about 75 db/km, in other words, we need about half of the short wavelength solar energy needs to reach the very bottom of the glass melt. As the figure below shows, Rayleigh scattering only rises that level in the ultraviolet.

Spectral attenuation of glass fiber materials showing short-wavelength Rayleigh scattering for silica. Quoted from Giusy Origlio, "Properties and Radiation Response of Optical Fibers: Role of Dopants." 

Attenuation due to metallic ion impurities in optical fibers. Quoted from Murata, "Handbook of Optical Fibers and Cables."

The chart above suggests that a concentration of about 60 ppb of Fe2+ could be tolerated in a glass-melt used for solar energy storage. That level of purity, 60 ppb, is still about 300 times the purity of the 20 ppm silica that is currently available at a price affordable for use as thermal storage.

Absorption by iron depends on the oxidation state of the ion, the ferrous, Fe2+, or the ferric, Fe3+. It appears that the more oxidized state, Fe3+, might be preferable for solar energy storage, because of lower absorption. Quoting from Bahman Mirhadi and Behzad Mehdikhani,"Effect of Batch Melting Temperature and Raw Material on Iron Redox State in Sodium Silicate Glasses":

Because iron has two redox states, Fe2+ and Fe3+, and these states have different colors, control of the redox condition during melting is important in adjusting the transmittance of glass in the wavelength regions of UV, visible light, and IR. When melted glass is placed in a reductive environment, the content of Fe2+ increases, and the glass turns bluish green because Fe2+ has a broad absorption band centered at 1050 nm. This absorption band is utilized in IR-cut glass such as cold-filtered glass which is produced under reductive conditions. When melted glass is placed in an oxidative environment, the iron in the glass is oxidized to the Fe3+ state and produces a yellowish color due to an absorption band which shows strong absorption below 380 nm that tails off towards longer wavelengths up to about 450 nm.

Mirhadi and Mehdikhani found that the redox ratio, Fe+2/Fe3+, increases in the solidified glass with melt temperature, reaching a plateau at about 0.3. 

Mirhadi and Mehdikhani found that the redox ratio, Fe+2/Fe3+, increases in the solidified glass with melt temperature, reaching a plateau at about 0.3. If Fe3+ is indeed strongly preferable, then 180 ppb of total iron may be acceptable in a glass melt used for solar energy storage since only a third will be in the  more absorbing Fe2+ redox state. That is still 100 times the purity of silica currently available at an acceptable price.