Solar Irradiation and Energy from Deserts

Steve Herron
November 28, 2010

Submitted as coursework for Physics 240, Stanford University, Fall 2010


The enormous amount of energy provided by the sun makes solar energy a very attractive alternative energy source. The sun constantly delivers about 120,000 terawatts (TW) of power to the earth, which is approximately 4000 times the entire global projected energy demand by 2050 of 26.4 to 32.9 TW (including both as electricity and fuels). [1,2] Harnessing significant amounts of this energy requires large areas of land with high insolation, or amount of solar irradiation. Deserts have become an attractive site for solar power plants, possessing both bountiful year-round insolation and land that does not compete with agriculture or civilization. The world's largest initiative to harness solar power from deserts is the organization known as DESERTEC, which currently is endorsing use of the Sahara Desert to power Europe, the Middle East, and Northern Africa (EU-MENA) with a large fraction of their electricity by 2050. [3] One of their major claims is: "A concentrating solar collector field with the size of Lake Nasser could harvest energy equivalent to the present Middle East oil production." [4] The purpose of this article is to construct the framework for addressing this claim by examining the energy available in solar radiation as well as solar energy technologies and their conversion efficiencies.

Solar Irradiation:

How much energy is available from the sun? Outside of the atmosphere, the sun provides the earth with an average direct flux of [5]:

Three main factors decrease the amount that reaches the ground: (1) absorption and scattering by air, (2) exposure at angles other than direct sunlight, and (3) weather patterns. The first effect can be accounted for by assuming a linear absorption of light by the atmosphere:

where α is the extinction coefficient of the atmosphere and η is the airmass, or path length of air between the earth and sun. While a good model of airmass for an angle θ around 90° from the horizon is

it reaches a singularity at the horizon and moreover does not account for the curvature of the earth. A more accurate and usable model of airmass as a function of angle has been proposed by Pickering: [6]

where θ is again the angle from the horizon in degrees. The average irradiation normal to the ground at sea-level at the equator can then be calculated:

where sin(θ) accounts for the path of the sun normal to the ground (this is equivalent to the light solar panels that do not track the sun would absorb) and 180 is added for normalization purposes, and the factor of two averages the exposure over twelve hours in the day and twelve in the night. The extinction coefficient α can vary significantly depending on the weather. For a dry, sunny location, a typical value for stellar absorption of 0.20 can be used although it should be noted that α can change dramatically with weather conditions. [7] We obtain

Additional losses at latitudes other than the equator's can be estimated by multiplying by the term cos(φ) where φ is the latitude of the location. Furthermore, weather patterns must be taken into effect. As an example, we can look at Las Vegas, Nevada, with a latitude of φ=36.08°N and a climate that is reported to receive 85.0 percent of the possible sunshine. [8]

This value is in close agreement with the 1960-1991 average observed value in Las Vegas of 5.71 kWh/m2 per day for a stationary, flat collector with no tilt. [9] Some solar energy technologies employ collectors that rotate on a North-South axis to track the sun. The above model can be modified to approximate the solar radiation for a one-axis tracking system by removing the sin(θ) term:

This is also in relative agreement with the reported value of 8.1 kWh/m2 per day for a flat-plate tracking collector with a North-South axis and no tilt. [9] Note that while the irradiance on the tracking panels increased from those stationary, the total irradiance of the land they occupy cannot increase over the former value.

Solar Conversion Technologies:

The two major technologies for converting solar energy to electricity are photovoltaics (PV), which rely on the creation and selective separation of electron-hole pairs through photon absorption, and solar thermal devices, which concentrate sunlight with mirrors to heat a fluid to high temperatures for operation in a thermodynamic engine. (Solar thermal is often referred to as concentrated solar power (CSP), although some photovoltaics operate under concentration as well.) The cell efficiencies of photovoltaics can range from 15% to over 40% between various technologies, although the efficiency of modules for large-area production is lower: First Solar claims their CdTe modules have reached an efficiency of 11.3%. [10]

Similarly, there are several types of CSP with varying efficiencies: dishes, troughs, towers, and Fresnel reflectors. The most efficient CSP technology is the concentrating dish system, where a parabolic dish tracks the sun on two axes, concentrates sunlight to a collector that heats to around 800°C, and powers a Stirling Engine at up to 31% sunlight to electricity efficiency. [11] Net annual efficiencies are estimated to reach from 12% to as high as 25% depending on the scale and technology. [12] To produce large amounts of power, these dishes are connected in parallel, but can also be used in satellite locations for small localized power. The remaining CSP technologies can vary in efficiency by plant size, as they concentrate the solar energy over the entire site to a single source where it is limited by the efficiency of the Rankine Cycle.

Parabolic troughs track the sun on one axis (usually North-South) and focus light on a pipe running through the focal point. Fresnel reflectors are similar to parabolic troughs but employ flat mirrors to focus the light, and will not be discussed further. The most extensive parabolic trough project was called Solar Energy Generating Systems (SEGS) in the Mojave Desert, California, where nine large plants were constructed and tested. The sixth one, SEGS VI, achieved reported sunlight to electricity efficiencies of 10.6%, while modern combined-cycle plants can achieve efficiencies of 16%. [12]

Power towers utilize heliostats that track the sun on two axes and focus light on a single collector on a centrally-located tower. Notable tower projects have been Solar I and Solar II (now decommissioned) in California, PS-10, PS-20, and the planned Solar Tres in Spain. Power towers have peak efficiencies of 23% and net annual efficiencies of an estimated 7-20% again based on the scale and technology. [12] Towers and troughs have the significant advantage over both photovoltaics and dish concentrators in that they can store thermal energy in molten salt and use it when the sun is not shining to still produce power. The capacity factor of a power system is defined as the average power produced throughout a 24-hour cycle divided by the output at full capacity and is a measure of how well power can be provided throughout the day. The capacity factors of troughs, towers, and dishes without any form of storage are all limited to 20-25% at best. With storage, parabolic troughs and power towers can reach an estimated 50% and 77%, respectively, making them great candidates for all-solar power generation plants. Molten salt storage can be as high as 99% efficient in round-trip storage, and will therefore incur no losses on overall energy production of a solar plant. [12] Adding molten salt storage to a solar plant would therefore have no effect on the overall energy production while capacity factor would rise and nominal capacity would shrink proportionally.

Estimating Overall Efficiencies

In making estimates for the real efficiency of large scale solar plants and estimating the land required for a certain power generation, the nominal capacity, capacity factor, and the total land use of a solar plant must be taken into account to make a real assessment. I will look at three examples of modern solar power installments and assess their overall real efficiency over the land they occupy: (1) Nevada Solar One, a parabolic trough plant in Boulder City, NV, (2) Copper Mountain Solar, a photovoltaic plant across the street from Nevada Solar One, and (3) PS-10, the solar power tower near Seville, Spain.

Nevada Solar One

This parabolic trough plant owned by the Acciona energy company is rated at 64MW, does not contain a storage mechanism, and is listed as a project site of 1.6 km 2.[13] The solar collectors are designed, as claimed by Solar Electric Power Association, to produce 134 GWh/year, which would equate to a reasonable capacity factor for a plant without storage of 21%. [14] Therefore, the total average power density would be

From the previous calculation for the irradiation of Las Vegas Nevada of 224.5 W/m2, Nevada Solar One's overall efficiency would be

Copper Mountain Solar

Copper Mountain Solar is a photovoltaic plant owned by the Sempra Energy company that uses First Solar's newest CdTe modules to supplement their 480 MW natural gas plant in Boulder City, NV. The photovoltaic plant started at 10 MW called El Dorado and the expansion to a total 58MW was recently completed. [15,16] El Dorado covers about 80 acres and Copper Mountain covers 380 acres, giving the total 58 MW plant an area of 460 acres, or 1.86 km2. [16,17] We can assume a capacity factor of 26.3% for these photovoltaic plants, a typical value for high-insolation areas [12] Copper Mountain Solar's overall average power density is therefore

The overall efficiency would be

PS-10 Solar Tower

PS-10 is an 11 MW Solar Tower owned by Abengoa with limited energy storage capacity (less than an hour's worth) that can produce a claimed 24.3 G Wh/year. [18] This equates to a capacity factor of 25.2%. The reported value of PS-10's total land I encountered is 0.55 km2, which makes the total average power density: [19]

The average insolation in Seville, Spain is 4.85 kWh/m2 per day, or 202 W/m2. [20] The PS-10 overall efficiency is calculated to be

The efficiencies are summarized in Table 1.

Solar Plant: Nevada Solar One Copper Mountain Solar PS-10
Overall Efficiency: 4.3% 3.7% 2.5%
Technology: CSP - Trough PV - CdTe CSP - Tower
Table 1: Overall efficiencies of modern large scale solar plants.

Addressing the Claim

The overall efficiencies calculated for the two solar thermal technologies were 4.3% and 2.5%, while the photovoltaic plant was between that range at 3.6%. In addressing the DESERTEC claim, we can therefore assume that the realistic efficiency of large scale solar operations would be about 3.5% if a combination of the above technologies were used. The Middle East oil production is listed at 1156.4 Mtonnes/year for 2009. [21] Using the conversion factor of 43.8 MJ/kg, the total power of Middle East oil production is:

For the average insolation in the Sahara, we can use data from the latitude and longitude of Aswan, Egypt, a city at the north end of Lake Nasser, which is reported to receive an average of 263 W/m 2 throughout the year. [21] Using DESERTEC's area of Lake Nasser of 6000 km2, and the above average efficiency of 3.5%, we can calculate how many Lake Nassers of solar plants it would take to match the Middle East oil production:


By the above calculations, the DESERTEC claim underestimates the area a solar collector field would require to match the Middle East's oil production by a factor of 30. Instead, it appears that one Lake Nasser could only match that supply only if 100% of the solar energy on the entire area were converted to electricity. However, this result remains very encouraging for the DESERTEC initiative: The Sahara desert covers approximately 9.4 million km2, and covering less than 2% of it with 3.5% overall-efficiency solar power plants would surpass the energy content of Middle East oil production. From a physical standpoint, the energy is indeed there.

© 2010 Steven M. Herron. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


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