## Solar Power Technical Notes

by Wm. Robert Johnston
last updated 14 February 2005

This page provides documentation and additional information regarding the analysis in Solar Power: An Exercise.

• Question 1: The solar constant of 1,368 W/m2 is for the full solar spectrum above the Earth's atmosphere at the Earth's mean distance from the Sun. This is a mid-range value; mean values measured by different satellites vary due to differing calibrations [1]. Additionally, solar irradiance varies about this value by about ±1.5 W/m2 over the 11-year solar cycle.

Some additional energy conversions which may be useful:
1 petajoule (PJ) = 1015 joules = 278,000 megawatt-hours (MWh) = 23,900 tonnes of oil equivalent (toe) = 9.48 x 1011 British thermal units (Btu)

• Question 2: Sources differ slightly on data regarding the Earth's radiation balance. The figure used here of 22% prevented from reaching the ground by the "clear" atmosphere (i.e. in the absence of clouds) includes 6% reflected into space and 16% absorbed. [2] Globally, another 23% on average does not reach the ground due to reflection or absorption by clouds, but this will be ignored here.

• Question 3: A latitude of 26° corresponds to Brownsville, Texas, or Miami, Florida. A latitude of 41.2° corresponds to the average latitude for the ten largest metropolitan areas of the U.S. northeast and midwest. For the winter solstice, the incident angle of sunlight will be the latitude plus 23.5°, the Earth's axial tilt.

• Question 4: Note that the calculation is correct for flux at the equinoxes only, when sunlight is incident perpendicular to the Earth's axis. The actual annual average requires additional geometric considerations. The factor of 0.8 for winter daylight in the northern U.S. [3] is only an approximate correction, due to the same geometric considerations.

Various sources [4, 5] provide maps of average solar flux, accounting for geometric factors, variable atmospheric absorption, and variable cloud cover. Using data from the National Renewable Energy Laboratory (NREL) [5, 6] the following approximate values are obtained:

• South Texas annual average, 200 W/m2;
• northern U.S. annual average, 140 W/m2; and
• northern U.S. December average, 71 W/m2.

Comparing to the values obtained in question 4, the derived additional weather/geometric factors are 0.66, 0.55, and 0.62, respectively.

The same NREL data [6] indicates that the best annual average solar flux in the United States, which is for locations in the southwestern U.S., is about 240 W/m2. Average annual flux reaches 250 W/m2 in parts of Australia, about 300 W/m2 in Sudan, but 150 W/m2 is representative of Europe, Japan, and the Northeastern U.S. [23]

• Question 5: The U.S. Department of Energy (DOE) reports the best demonstrated efficiencies for photovoltaic cells as 24.7% for crystalline silicon cells [7] and 19.2% for one type of thin film cell. [8] Nelson reports a 25.1% efficiency for monocrystalline gallinium arsenide cells. [23] A report for the NREL cites a 30.3% efficiency for a multijunction thin film cell. [9] A Boeing article reports an efficiency of 34% for a multijunction cell [10], although this cell (and others attaining similar efficiencies) involve concentration of sunlight within the cell.

Commercial cell efficiencies lag behind these values, since mass production requires the development of economical means of production. The U.S. DOE cites 16% as representing the best commercially-available crystalline silicon cells. [7] The U.S. Energy Information Agency (EIA) reports efficiencies of in-use single-crystal cells as 14-16% in 1998. [11]

Physical limitations prevent the production of perfectly efficient photovoltaic (PV) cells. The individual photons in sunlight can only liberate electrons in PV cells if they have energies within a certain range, this range being characteristic to the materials and construction of the cell. In the case of thin-film cells, the thermodynamic limit for single-junction photovoltaic cells is an efficiency of 30%, and for an infinite number of junctions is a 68% efficiency. [9] Concentration of sunlight can improve efficiency of single-junction cells to 37% to 40%, although the concentration factors required are significant. [23] By comparision, the theoretical limit for efficiency of crystal silicon cells is about 28% [12] or 29% [23] versus about 31% for gallinium arsenide cells [23].

These efficiencies for solar photovoltaic cells are one reason why most solar energy production today is accomplished with solar thermal systems. The theoretical efficiency limits cited above do not apply to the production of heat from solar thermal systems. However, solar thermal systems face the same current efficiency in conversion to electrical energy as other thermal electric plants: about 35%. [13] In fact, for any type of energy conversion from sunlight (be it solar cell-based or solar thermal-based), thermodynamics imposes a maximum possible efficiency of 85%. [23]

• Question 6: The U.S. EIA gives U.S. residential energy consumption in 2003 as 21,229 trillion BTU. [14] This figure is primary energy, however, and includes thermal waste energy in central electricity generation. If solar photovoltaic generation is to replace residential energy needs, it must cover the 7,242 trillion BTU of primary energy consumption (mostly fuels burned within the residence) plus the 4,367 trillion BTU of electricity used in the residences. The resulting total of 11,609 trillion BTU excludes the energy lost in central generating facilities and in transmission to residences. This is an average of 388,600 MW for the year. The U.S. Census Bureau estimated number of households in 2003 is 120,879,390 [15] which yields an average household energy use of 3,215 W.

• Question 7: The 18-kW power requirement for a compact car is from Petr Beckmann. [16] UMR gives a requirement of 15 kW. [17]

• Question 8: Battery efficiency (for extraction of input energy) is assumed to be 80% (some sources [17] suggest lower values).

• Question 9: For 2003, the U.S. EIA reports total U.S. electric power generation as 3,883,185,000 MWh, imports of 30,390,000 MWh, and exports of 23,972,000 MWh. This yields a consumption of 3,889,603,000 MWh, or an annual average power consumption of 444,000 MW. [18]

• Question 10: The U.S. EIA reports total U.S. energy consumption in 2003 as 98,156 trillion BTU [19], for an average during the year of 3.29x106 MW. This is primary energy.

(The result from this question--42,700 km2--may be compared to an estimate by the U.S. DOE which corresponds to 39,700 km2 [20].)

• Question 11: The U.S. EIA reports total world energy consumption in 2002 as 411.57 quadrillion BTU [21], for an average during the year of 1.38x107 MW. This is primary energy.

• Question 12: The United Nations published new population projections in 2004 [22], including a median projection with the stabilizing of world population after reaching a peak of 9.22 billion around 2075. The same report estimates world population in 2005 at 6.45 billion.

Note: this exercise is oriented towards solar power, and specifically towards photovoltaic solar cells (although much of this analysis can be applied to other methods of solar power generation). For information on other types of renewable energy and further information on solar energy, see the Environmental Topics main page.

Sources:

© 2001, 2005 by Wm. Robert Johnston.
Last modified 14 February 2005.
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