by Vaclav Smil
Power Density: Sorting Out the Rates
There is no single, binding, universal definition of power density as different science fields and different branches of engineering-including electrochemistry, telecommunications, and nuclear electricity generation-have used the term for a variety of kindred but distinct rates, with mass, volume, and area as denominators. Then there is a revealing, and virtually universal, notion of power density as the energy flux in a material medium, a concept that can be used to assess the potential performance of all modern commercial energy conversions. And in this book I relate energy flux to its fundamental spatial dimension by quantifying power that is received and converted (or that is potentially convertible) per unit of land, or water, surface. Before I begin doing so I will briefly review the other applications of power densities.
In electrochemistry, power densities, expressed both in volume (W/cm3) and in mass (W/g) terms, are used to rate the performance of batteries. While energy density Q/g) measures the specific energy a battery can hold (a higher rate implying, obviously, a longer run time), power density measures the maximum energy flux that can be delivered on demand in the short bursts of electricity required for tools, medical devices, and in transportation. Early lead acid batteries delivered less than 50 W/kg, but by the end of the twentieth century the power densities of these massively deployed (above all in cars) units were between 150 and 300 W/kg. Utilitygrade batteries can deliver up to 40 MWh of electricity with efficiencies of up to 80%, and their power densities range between 200 and 400 W/kg at an 80% charge level. Since the year 2000 the best performance of experimental lead acid batteries has been boosted by the addition of high-surfacearea carbon: its addition adds only up to 3% of weight but increases surface area by more than 80%, to yield a power density of more than 500 W/kg; the ultimate target is 800 W/kg (Svenson 2011).
More expensive nickel-cadmium batteries can supply up to about 1 kW/kg, and the increasingly common lithium-ion batteries deliver well in excess of that (Omar et al. 2012; Rosenkranz, Kohler, and Liska 2011). Highenergy Li-ion batteries deliver more than 160 Wh/kg, but their power density is less than 10 W/kg; in contrast, very high-power Li-ion batteries, commonly used in electric vehicles (where they have to deliver more than 40 kW), rate less than 80 Wh/kg but have power densities up to 2.4 kW/kg at an 80% charge; as the density of these batteries is about 2.1 kg/L, that translates to almost 5 kW/L. By 2014 the best commercially available rates were about 2.8 kW/kg, and the world's highest-power-density Li-ion car battery, a prismatic cell revealed by Hitachi in 2009, can discharge as much as 4.5 kW/kg (Hitachi 2009).
In nuclear engineering, the average core power density is the amount of energy generated by the specific volume of the reactor core, a quotient of the rated thermal reactor power and the volume of the core (IAEA 2007). Its value is usually expressed in kW/dm3 (kW/L), and the range for all reactors listed in the International Atomic Energy Agency's Prins Database is 1-150 kW/dm3. Early British Magnox reactors and advanced gas reactors (AGRs)-designs that used graphite moderator and CO2 cooling when they pioneered the UK's commercial fission electricity generation during the late 1950s-had a power density of, respectively, 0.9 and 3 kW/dm3. The rates for pressurized water reactors (PWRs, water-cooled and water-moderated), the dominant choice for commercial nuclear electricity generation around the world, are mostly between 70 and 110 kW/dm3. The highest rates (in excess of 700 kW/dm3) were reached in experimental fast breeder reactors cooled by molten salt (Zebroski and Levenson 1976; IAEA 2007). Power densities in traveling-wave reactors, now under development, would be about 200 kW/dm3 within the active fission zone.
Telecommunication engineers routinely calculate the power density for energy received from transmissions emanating from both isotropic and directional antennas. One of the main reasons for this is to make sure that human exposures to nonionizing radiation do not exceed accepted safety standards. The US Federal Communications Commission (FCC) set the maximum permissible exposure for power density for transmitters operating at frequencies between 300 kHz and 100 GHz (FCC 1996). Between 30 and 300 MHz (very high-frequency wavelengths of 1-10 m that carry FM radio and television broadcasts) the limit is 1 mW/cm2 for occupational exposures averaging 6 minutes, and just 0.2 mW/cm2 for the general population and uncontrolled exposure averaging 30 minutes.
For shortwave broadcasts (frequencies of 2.3-26.1 MHz), the general population limit in mW/cm2 is 180/f2: this means that an international BBC broadcast at 15 MHz would allow for the maximum exposure of 0.8 mW/cm2. Typical shortwave radio transmitters have a power of 50-500 kW, while many longwave transmitters rate more than 500 kW, and the world's most powerful one requires 2.5 MW. The power density (PD) of an isotropic antenna (radiating energy equally in all directions) is simply a quotient of the transmitted power (Pt, peak or average) and the surface area of a sphere at a given distance: PD = Pt/4ir2. A 100-kW transmitter would thus produce a PD of 0.8 nW/m2 at a distance of 1,000 km, equal to only one-millionth of the allowable exposure.
In reality, most radio antennas have considerable transmission gain (Ge), which is created by suppressing upward and downward directions and concentrating the output toward the horizontal plane. After correcting for this intervention (PD = P,Gt/4itr2), a 100-kW shortwave transmitter with a gain factor of 10 will have an effective radiated power of 1 MW and a PD of 8 nW/m2 at 1,000 km. Shortwave broadcasting antennas rely on particularly narrow beam widths in order to transmit their signal between continents; so do, of course, radar antennas that require a high gain of up to 30 or 40 dB (that is, a Gr between 1,000 and 10,000) in order to pinpoint distant targets (Radartutorial 2014).
For the ultra-high frequencies (1.9 GHz) used by cell phones, the general population limit is 1 mW/cm2, while the actual received maxima near a cell phone tower are 10 pW/cm2, that is, just 0.01 mW/cm2. Higher power density exposures apply to time-varying electric, magnetic, and electromagnetic fields between 30 and 300 GHz; these wavelengths of 1-10 mm are the highest radio frequency just below the infrared radiation. The International Commission on Non-Ionizing Radiation Protection puts the maximum power densities at 50 W/m2 for occupational exposure and 10 W/m2 for general public exposures (ICNIRP 1998).
Umov-Poynting Vector
Electrochemists, reactor physicists, and radio engineers thus use power densities with three different denominators (mass, volume, and area), but a universal approach makes it possible to assess the power density of virtually all energy conversions by quantifying the energy flux per unit area of the converter's surface. In 1884 John Henry Poynting (1852-1914; fig. 2.1), a professor of physics at the University of Birmingham, set out to prove
that there is a general law for the transfer of energy, according to which it moves at any point perpendicularly to the plane containing the lines of electric force and magnetic force, and that the amount crossing unit of area per second of this plane is equal to the product of the intensities of the two forces, multiplied by the sine of the angle between them, divided by 4 it while the direction of flow of energy is that in which a right-handed screw would move if turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. After the investigation of the general law several applications will be given to show how the energy moves in the neighbourhood of various current-bearing circuits. (Poynting 1884, 344)
This directional transfer of energy per unit area (energy flux density, measured in W/m2) became known as the Poynting vector, but, as is often the case in scientific discovery, a Russian physicist, Nikolai Alekseevich Umov (1846-1915; fig. 2.1), formulated the same concept a decade earlier (Umov 1874), and hence in Russia the measure has been known as the Umov-Poynting vector. Piotr Leonidovich Kapitsa (1894-1984), Ernest Rutherford's student and Nobel Prize winner in physics (in 1978, for his basic discoveries in the area of low-temperature physics), pointed out that the vector can be used to assess all energy conversions to reveal the "particular restrictions of these various flows," which ar
e often ignored, resulting "in wasting money on projects that can promise nothing in the future" (Kapitsa 1976, 10).
Figure 2.1
John Henry Poynting (left) and Nikolay Alekseevich Umov, the two originators of the concept of energy flux density. Stipple portraits by Noli Novak.
The Umov-Poynting vector thus offers a fundamental assessment of energy converters in all cases where
the density of the energy influx is limited by the physical properties of the medium through which it flows. The rate at which energy can be made to flow in a material medium is restricted by the velocity (v) of propagation of some disturbance (a mechanical wave or heat flow, for example) and the energy density (U) of the disturbance. The rate of flow (W) is always in a particular direction (it is a vector, like an arrow). Vector W is equal to vector v times U and proves very convenient for studying processes of energy transformations. (Kapitsa 1976, 10)
Of course, the final value must be multiplied by appropriate factors in order to account for maximum efficiencies. This is well illustrated by looking at electricity generation by a large modern wind turbine. Its 50-m-long blades will sweep an area A of roughly 7,854 m2; with a wind speed v of 12 m/s and an air density (at 20°C) of 1.2 kg/m3, the kinetic energy density (U= 0.5mv3) will be 1,037 J/m3, and the maximum power of the machine would be 8.14 MW.
Box 2.1
Maximum power of a wind turbine
This result must be corrected for the theoretical maximum efficiency: Betz (1926) established that its limit is 16/27 (0.59) of the potential. Multiplying 8.14 MW by 0.59 sets the maximum turbine power at 4.8 MW, but the actual performance is considerably lower because of unavoidable energy losses (in gearing, bearings), and the correction factors range between 0.35 and 0.45 even for the best-designed modern wind turbines. I chose the blade radius of 50 m because it matches that of the GE 2.5-MW series turbine (GE Power & Water 2010; fig. 2.2). The actual power coefficient of this large machine is only 0.3, and the effective power density of its electricity generation is 310.9 W/m2 of the area swept by its blades.
The vector can be used to find the limits of electric generators or combustion engines and, as Kapitsa noted at the outset of his paper, to refute some apparently appealing proposals: he related how he was asked to disprove the idea suggested by his teacher, the famous physicist Abram Fedorovich loffe, to use electrostatic rather than electromagnetic generators for large-scale electricity production. Electrostatic generators would be easier to build and could feed high voltage directly into the electricity grid. But to avoid sparking, the electrostatic field is restricted by air's dielectric strength, and to generate 100 MW (a rate sufficient to supply the electricity needs of nearly 80,000 average American consumers), the electrostatic rotor would need to have an area of about 400,000 m2 (nearly 0.5 km2), obviously an impossible requirement.
Figure 2.2
GE 2.5-MW wind turbines. Photo reproduced by permission.
The maximum power that can be transmitted during combustion from a burning medium to the working surface (an engine piston or rotating turbine blades) is the product of gas pressure, the square root of its temperature, and a constant dependent on the molecular composition of the gas. The vector also makes it clear why some very efficient energy conversions are not suitable for high-power supply because of the low power densities. Fuel cells are an excellent example of such limitations: their peak theoretical efficiency of transforming chemical energy into electricity is about 83%, but low diffusion rates in electrolytes limit their power density to about 1 W/cm2 of the electrode.
This means that the working surface of fuel cells delivering 1 GW (a rate easily needed by a large city) would have to be on the order of 100,000 m2. Obviously, the power density of fuel cells is too low to provide the centralized base-load supply for modern urban, high-energy settings (Brandon and Thompsett 2005). In contrast, in modern large thermal turbogenerators rated at 1 GW (enough to provide electricity for at least 750,000 average US consumers) the high velocities and high temperatures of the working medium (steam superheated to 600°C, traveling at 100 m/s with a density of 87.4 kg/m3 at 30 MPa) create power densities as high as 275 MW/m2 across the area swept by the longest set of blades rotating at 3,600 rpm. I return to Umov-Poynting vector densities when reviewing the performances of some modern energy converters.
My final example of the prevailing lack of consensus regarding the application of power density can be found by consulting Elsevier's six-volume Encyclopedia of Energy (Cleveland et al. 2004). Four authors use the measure in four different ways. In the first volume, Michael M.Thackeray (2004, 127), reviewing batteries for transportation, defines the rate as "power per unit of volume, usually expressed in in watts per liter." In the third volume, J.M.German (2004, 197), writing about hybrid electric vehicles, employs mass denominator, "the power delivered per unit of weight" (W/kg). I use it, also in the third volume, when looking at land requirements of energy systems (Smil 2004b, 613), as "average long-term power flux per unit of land area, usually expressed in watts per squared meter." And in the sixth volume, Arnulf Griibler (2004, 163), writing about transitions in energy use, defines it as the "amount of energy harnessed, transformed or used per unit area."
W/m2
As already noted, my primary measure of power density in this book is not the energy flux per unit of the working surface (vertical or horizontal) of a converter (as in the Umov-Poynting vector) but per unit of the Earth's surface, for comparability always expressed in W/m2. I have been advocating the use of this measure since the early 1980s (Smil 1984) and chose it as key analytical variable in my first synthesis of general energetics (Smil 1991) and again in its thoroughly revised and expanded sequel (Smil 2008). Since the late 1990s, power density, expressed as energy flux per time per unit of horizontal surface, has been receiving more attention as a result of the growing interest in renewable energy resources and their commercial conversions to fuels and electricity. The power densities realized by harnessing these energy flows are appreciably lower than the power densities of fossil fuel-based systems, something that even often uncritical proponents of wind and solar energy-the new renewables, in addition to the two traditional renewable energies, hydro energy and fuelwood-cannot ignore.
Perhaps the greatest advantage of power density is its universal applicability: the rate can be used to evaluate and compare all energy fluxes in nature and in any society. In this book I use it for a systematic assessment of all important energies, be they natural (renewable) flows or fossil fuels burned to produce heat for many direct uses or as the means of generating thermal electricity. Because fossil fuels are the dominant source of primary energy, I look first at the power densities associated with producing, processing, and transporting coal, crude oil, and natural gas, and then at combustion processes in general and at the thermal generation of electricity in particular. A significant share of the latter activity is also energized by uranium fission, and I quantify the power densities of the entire nuclear fuel cycle.
I also quantify, in a brief historical survey and in more detailed sectoral comparisons, the hierarchy of power densities of energy uses and the challenges of heat rejection arising from highly concentrated energy conversions. In the penultimate chapter I summarize the book's main findings to demonstrate why power densities matter and why their appreciation and assessments should be among the key ingredients of any attempts to understand the energy predicament of modern, high-energy civilization and to guide the development of its future energy supplies. Although there is no imminent global prospect of running out of fossil fuels (the very notion of running out as a complete physical absence of a resource is incorrect, as rising prices will end an extraction long before the last reserves are exhausted), such resources are finite, and eventually our civilization will have to make the transition to renewable energy sources.
The early stages of this transition have been unfolding during the past generation, and in the book's last chapter I explain some of the challenges involv
ed in shifting from a high-power-density system of energy supply based largely on fossil fuels to low-power-density renewable energy flows and their commercial energy conversions to produce electricity as well as substitutes for oil-based liquid fuels. But before starting a systematic review of specific power densities I should take a closer look at the qualitative differences that are hidden by a simple quantitative expression. They affect the rate's numerator: all energies can be converted to a common denominator, but that measure cannot convey their qualities. And the qualities are no less important as far as the denominator is concerned: space attributable to energy production and use may be easily measured in common units, but its qualitative attributes range across a very wide spectrum of land cover transformations.
Typology and Caveats
Power density is an inherent property of all forms of energy production and energy use. These processes unfold on scales ranging from subatomic to universal, but we are interested above all in their varied and ubiquitous commercial manifestations that sustain our civilization. Power density, however, is not a primary design parameter. The energy systems of modern societies are designed for overall performance and reliability achieved at acceptable cost: annual output (and hence maximum available power), price per unit of product, and desired parameters of safety and environmental impacts govern their construction and performance, and a specific power density is simply a key consequence of their operation.