Truth in labeling: Solar multiple vs. capacity factor

It is natural to discuss a solar plant in terms of its rated output (i.e., peak electrical power,) but this often gives a false impression of the plant's economic value. For example, a utility might expect a new coal-fired plant to annually generate electric energy equivalent to its producing at full power 20 hours per day (capacity factor = 20/24 = 83%.) A solar plant in a desert climate will, over the course of a year, only produce energy equivalent to its producing at full power about 6 hours per day (capacity factor = 6/24 = 25%)

With energy storage a solar plant can operate with a higher capacity factor (saving money on some aspects of the plant) and making its rated output more directly comparable with a conventional plant. In a simplistic calculation, a 6 hr/day solar plant can become an 18 hr/day solar plant if we divert 2/3 of its output to storage. That is, by later withdrawing the stored energy we can have two more 6 hr periods of full power operation, giving a total of 18 hr/day. Our hypothetical solar plant would be said to have a capacity factor of 75%, 12 hours of storage, and a solar multiple of 3 (i.e., the ratio of its rated power without storage to its rated power with storage.)

In the real world, the above calculation would require a computer simulation, but we can borrow some real world numbers from Gemasolar, a thermal solar plant in Spain, which has 15 hours of storage and a 75% capacity factor. For obvious reasons, solar plants with a significant amount of storage prefer to advertise their annual electricity production rather than their rated power. From Gemasolar's stated annual production of 110,000 MWh/yr, and stated capacity factor of 75%, we can calculate that its rated power is:

110,000 MWhr/yr * (1/0.75) * (1/8760) yr/hr = 16.7 MW

Gemasolar has 304,750 m² of mirrors, so its rated power comes to 55 w_e/m² of mirror area.

Gemasolar and the Ivanpah thermal solar plant differ in detail (and Ivapah has the better solar climate,) but it is relevant to note that Ivanpah has three units rated 123 MW + 133 MW + 133 MW = 389 MW total, no storage, and 1,079,232 MWh/yr annual output—a capacity factor of 32%. In total, Ivanpah has 2,600,000 m² of mirror area, so its rated output of 389 MW comes to 150 W_e/m² of mirror area. That suggests that Gemasolar's solar multiple is about 150/55 = 2.7

The conclusion here is that solar plants without storage should be de-rated by approximately a factor of three before being directly compared with conventional fossil-fueled plants on the basis of rated power.

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,

Direct absorption and storage of solar energy in glass melts

In a glass-making solar furnace, solar energy is directly absorbed in the semi-transparent melt.

Contrary to popular belief, renewable power does not "need" energy storage. When a GW of wind or solar power is brought online, the electric utility's least fuel-efficient 1 GW of conventional generating capacity is forced into semi-retirement. That is, those particular generating plants no longer have a job when the wind is blowing or the sun is shining. Since we have about 4 TW of conventional generating capacity to semi-retire in this way, renewable power will not be hurting for energy storage anytime soon. 

That said, in a thermal power plant some energy storage comes free—or at least at no additional cost—in the form of thermal inertia. The larger the plant, the more running time is extended by thermal inertia—and the cheaper it is to deliberately increase. Any process served by a solar furnace may benefit from this inexpensive form of energy storage. Since an all-glass, glass-making solar furnace will be first and foremost occupied in making its own glass parts, it is reasonable to look at the thermal inertia in the glass melt itself. 

A 2002 paper by L. Pilon, G. Zhao, and R. Viskanta looked at the thermophysical properties of glass melts. A melt of soda-lime glass is substantially transparent to both sunlight and high-temperature thermal radiation, so molten glass effectively has high thermal conductivity when it absorbs solar radiation directly or cools radiatively from high temperatures. For example, at 1400 °C (1700 °K,) a soda-lime glass melt has an effective thermal conductivity (phonic conduction + radiation) of 58 W/m-°K—that's more than the thermal conductivity of steel at room temperature.

Pilon et al. also give some representative numbers for industrial glass-making. They considered a glass melting tank approximately 16 m long, 7 m wide, and 1 m deep heated from above with a total heat input of 8.3 MW which averages to 72 suns (i.e., kw/m2) over the free surface of the melt. They estimate a maximum heat flux of 134 suns near the center. At melt surface temperatures around 1500 °C (1800 °K) they associate the maximum flux with vertical temperature gradients of about 1200 °C/m. At about 8 MWth, such an industrial glass-making furnace is only a small-scale model of a GW-scale solar glass-making furnace.

A coal-fired furnace for a 800 MWe generating unit might be 20 m x 20 m x 100 m high, corresponding to an average thermal flux per unit wall area of about 250 suns.

T-s diagram for a supercritical power plant. According to L & T Power, typical temperatures at points E and G for current technology are 565°C and 593°C, respectively; efficiency = 42%.

Heat balance for an advanced 800 MW power plant. Image quoted from Song Wu et al., "Technology options for clean coal power generation with CO2 capture." Mean temperature in the first heat (596 + 293)/2 = 445°C; second heat (608 + 342)/2 = 475°C; efficiency = 46%.

The steam tubes absorb heat over a range of temperatures, but 460°C may be taken as representative for the advanced supercritical cycle in the diagram above. At 460°C, a blackbody radiator emits about 16 suns (20 kw/m2.) The molten glass will need to be significantly hotter at its "empty" temperature in order to transfer a flux 250 suns to the furnace's steam tubes (in order to drive operation at rated power.) If the product of the emissivities and the view factor is about 0.7, the glass melt must be at a temperature where a blackbody emits about 380 suns, that is, around 1340°C. Using the thermophysical properties of soda lime glass melts quoted in Pilon et al., and the modified Rayleigh number, Ra*, defined in Bolshov et al., 1340°C is well within the range of turbulent convection for soda-lime glass. If we assume the pool of molten glass is a hemisphere 100 m in diameter, and that the average volumetric heating rate is 30 kw/m3, we have Ra* = 3E13 when the glass melt is at 1340°C.

Thermophysical properties of molten glass as calculated from the relations in Pilon et al.

Convection flow patterns and isotherms in a hemispherical pool with isothermal walls and top. Image quoted from Bolshov et al. The modified Rayleigh number, Ra* = 1E8 above, and Ra* = 1E9 below—much lower than the Ra* = 3E13 estimated for a GW-scale energy store.

Observed turbulent convection at Ra = 6.8 E8. Image quoted from X. D. Shang, X. L. Qiu, P. Tong, and K.-Q. Xia, Phys. Rev. Lett. 90, 074501 (2003).