Radiative view factors in glass-melt solar storage

Radiative view factors between a cylinder and its endcaps. Diagram from Isidoro Martinez, "Radiative View Factors."

In radiative heat transfer, a view factor F_A→B is the proportion of the radiation which leaves surface A and (directly) strikes surface B. In some cases there may also be an adiabatic blackbody surface C which, at thermal equilibrium, must re-radiate all of the photons it receives. In this case we can define an effective view factor  F*_A→B that includes the re-radiated photons that reach B indirectly:

F*_A→B = F_A→B + (F_A→C)(F_C→B)

When all the surfaces are blackbodies (an assumption that greatly simplifies the math, and is approximately true in practice) the net radiative flux from A to B, averaged over  S_A, the area of A, is:

Φ_A = F*_A→B * σ(T_A⁴ - T_B⁴),

where σ is the Stefan–Boltzmann constant, σ = 5.7 E−8 W m−2 K−4.

The corresponding net flux at surface B, Φ_B, must be Φ_A multiplied by the ratio of the two areas:

Φ_B = Φ_A * (S_A/S_B) = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B).

When solar energy is stored in a glass melt, we can consider the free surface of the melt to be surface A (the lower cap of a cylinder,) the water wall tubes of the boiler to be surface B (the walls of the cylinder,) and the roof of the furnace to be the adiabatic surface C (the upper cap of the cylinder.)

Water wall tubes. Image quoted from http://tubeweld.com.

In the view factor diagram above, a reduced radius 'r' is related to the cylinder's radius, R, and its height, H, by:

r = R/H,

and a parameter ρ is defined to be:

ρ = (√(4 * r2 + 1) -1) / r.

The correspondences between our lettered surfaces and the diagram's numbered surfaces are:

A = 3, B = 1, C = 4.

From the diagram,

F_A→B = F_3→1 = ρ/2r

F_A→C = F_3→4 = 1 - ρ/2r

F_C→B = F_4→1 = F_3→1 = ρ/2r

So,

F*_A→B = F_A→B + (F_A→C)(F_C→B) =  ρ/2r + (1 - ρ/2r)(ρ/2r) = (ρ/2r) (2 - ρ/2r),

and the area ratio, S_A/S_B, is

S_A/S_B = πR² / 2πRH = R/2H = r/2.

The average flux on the water wall, Φ_B, is

Φ_B = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B) = (ρ/2r) (2 - ρ/2r) (r/2) * σ(T_A⁴ - T_B⁴)

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴)

With R = 51 m, H = 87 m, r = 0.58, ρ = 0.93, T_A = 1610 °K (1340 °C) , T_B = 730 °K (460 °C), as calculated in an earlier post,

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴) = 0.26 * 367 kw/m2 = 96 kw / m2

which is not nearly the 250 kW/m2 we need to see on a water tube wall.

We need to increase radiant transfer by upping the "empty" temperature of the glass melt. This will also incidentally decrease storage density, resulting in an increase in R and a decrease in H (as calculated on the basis of 250 kW/m2.)

A spreadsheet exploring radiant transfer in glass-melt thermal storage.

Exploring these relations in a Numbers spreadsheet shows that Tempty = 1570 °C gives a consistent solution with R = 63 m, H = 71, and the flux on the water wall tubes = 250 kw/m2. The glass temperature range from empty to full is just 230 °C.

Capacity factor and hours of solar storage

The Gemasolar plant in Andalucia, Spain operates at an annual capacity factor of 75% using just 15 hours of thermal storage.

The 17 MWe Gemasolar power tower in Fuentes de Andalucía, Spain is designed to operate at an annual capacity factor of 75%, and has run continuously for as long as 36 consecutive days. This remarkable accomplishment is achieved with just 15 hours of thermal storage. Clearly 15 hours of thermal storage is about the right amount for a solar plant!

Here are some statistics for Gemasolar gleaned from the National Renewable Energy Laboratory's site.

Projected annual output: 110,000 MWhr/yr = 12.6 MWe  annual average.
Rated output  (calculated from the 75% capacity factor): 16.7 MWe rated. 

Output per mirror area (304,750 m2) :
       41 We/m2 annual average,
       55 We/m2 rated.  

Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2-land annual average,
        8.6 We/m2-land rated

The 15 hours of storage based on 40% thermal efficiency is:

15 hrs * 3600 s/hr * 8.6 We/m2-land * 1/0.40 = 1.2 E6 Jthermal/m2-land

A plant with telescopic heliostats and glass-melt storage would have some advantages over Gemasolar. Telescopic heliostats can be packed much more closely, increasing the mirror/land ratio to around 0.70, increasing land yield about 4.5 times that of Gemasolar. Also, because the glass melt transfers its heat to hotter steam (608°C vs. 565°C) the steam cycle efficiency can be greater, about 46% thermal efficiency as compared with 40%, a factor of 1.15 .

Now, the same Gemasolar statistics if rebuilt on the same land with telescopic heliostats and glass-melt storage:

Projected annual output: 110,000 MWhr/yr  * 4.5 * 1.15 = 65 MWe  annual average.
Rated output  (calculated from the 75% capacity factor): 87 MWe rated. 

Output per mirror area (304,750 m2 * 4.5 = 1,370,000 m2) :
       41 We/m2 * 1.15 = 47 We/m2 annual average,
       55 We/m2 * 1.15 = 63 We/m2 rated. 

Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2 * 4.5 * 1.15 = 33 We/m2-land annual average,
        8.6 We/m2 * 4.5 * 1.15 = 45 We/m2-land rated.

The 15 hours of storage for the rebuilt plant becomes:

15 hrs * 3600 s/hr * 45 We/m2-land * 1/0.46 = 5.3 E6 Jthermal/m2-land

Plant Bowen in Euharlee, Georgia, is the largest coal-fired plant in the USA. It has four 800 MWe units, giving an aggregate rating of about 3.2 GWe. A telescopic heliostat / glass-melt power plant in Andalucia with 15 hours of thermal storage, having the same rated output of Plant Bowen, would occupy:

3.2 GWe-rated / 45 We/m2-land rated = 71 E6 m2,

or a circle 4.8 km in radius. The height of the central optics will be about 1/14 the field radius, or 340 m. This is about 11% higher than Plant Bowen's two 305 m smokestacks.

Plant Bowen, 3.2 GWe,