Power density



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Population density There are three kinds of commonly used density measures. The first, and undoubtedly the most commonly used rate, relates the number of individual items (be they organisms, people, or artefacts) to area. This spatial density is not among the SI derived units but the measure is common in ecological and population studies. In the first case densities of small organisms (plants, insects, invertebrates, small mammals) are measured as the number of individuals or their collective mass per m2, a more common rate for larger animals is individuals or total mass per hectare (ha) or km2, a measure that is also used for numbers and mass of trees, and for other phytomass (woodlands, grasslands, wetlands).

When population densities are expressed in relation to agricultural land the rate is almost always given as persons/ha, the rate ranging from about 1.25 ha/capita in Canada and 0.5 ha/capita in the US to less than 0.1 ha/capita in China and Bangladesh and to negligible areas (or even to nothing) in many small island and desert nations (FAO 2014). In agronomic studies density of planting or transplanting is also expressed per hectare: for example, corn densities in Iowa are now as high as 100,000 plants/ha but while yields initially increase with higher densities they level off and with no soil quality constraints optimum yields fall within the range of 70,000-80,000 plants/ha (Farnham 2001). For comparison, the highest-yielding hybrid rice varieties are grown with up to 200,000 plants/ha (Lin et al. 2009).

Comparisons of regional and national population densities are done as individuals/km2. This measure is particularly misleading as it prorates populations over entire national territories regardless their habitability: even in countries with relatively homogeneous population density it will hide large regional differences. For example, even in densely populated Netherlands it subsumes differences ranging over two orders of magnitude, from cities with more than 5,000 people/km2 to rural regions with fewer than 50 people/km2. Much larger differences in intranational population densities are common in all countries with very large territories as well as in nations located in arid and semi-arid climates, particularly in North Africa: Canada is an excellent example of these huge disparities among affluent countries, Egypt (with less than 4% of its territory under annual and permanent crops) offers the extreme example in the arid world. Moreover, continuing urbanization means that nationwide population densities are becoming universally less representative.

Density of human populations can be also expressed in mass terms (and care must be taken to specify the mass: either as live weight or in absolutely dry terms, the usage preferred in ecological studies). Most densely populated parts of the Asia’s still expanding megacities have residential densities on the order of 50,000 people/km2 and (using a conservative age/sex weighted mean of 45 kg/capita) they translate to live weight anthropomass of more than 2 kg/m2, the rate unmatched by any other mammal and three orders of magnitude higher than the peak seasonal zoomass of large herbivorous ungulates grazing on Africa’s richest grasslands (Smil 2013).

Average densities of health, commercial and recreational services (doctor offices, hospitals, food stores, children’s playgrounds, sports playing fields) offer a simple but revealing way to focus on spatial inequities and their consequences. For example, a study by Bonanno and Goetz (2012) revealed that even after controlling for missing variables, biases and lags, the density of stores selling fruits and vegetables (as opposed to many small establishment where only packaged and fast food is available) was associated with higher shares of adults who consumed fruits and vegetables regularly and had lower obesity rates.

Mass and energy density The second density category includes those derived SI units that relate variables to volume: mass density and energy density. SI mass density is measured in kg/m3 but in practice it is often expressed also in g/cm3, kg/dm3 or t/m3 (the number will be identical for these three rates). Densities of common materials (all expressed in g/cm3, with water as the yardstick at 1) range from 0.65-0.75 for most wood species to just short of 1 for plastics (polyethylene goes from 0.915 to 0.970 for its low- and high-density varieties), concrete has densities between 2.2-2.4, aluminum and its alloys are just above that at 2.6-2.7, steel alloys cluster mostly between 7.7-7.8 (Smil 2013a).

In SI energy density is a derived unit measured in J/m3 in SI but in energy publications this density is often express it (with the exception of gases) in mass terms as MJ/kg or GJ/t. This may be a cause for confusion because in SI nomenclature J/kg is a derived unit called specific energy. In SI units this specific energy is also often measured in J/g, MJ/kg or GJ/t. Accurate conversions between these two rates (from volume to mass or, in SI terms, from energy density to specific energy) require analyses of individual fuels. Energy density is one of the key determinants of the structure and dynamics of an energy system: there are many reasons to prefer sources of high energy density, particularly in modern societies demanding large and incessant flows of fuels and electricity.

Obviously, the higher the density of an energy resource the lower its transportation (as well as storage) costs, and this means that its production can take place further away from the centers of demand. Crude oil has, at the ambient pressure and temperature, the highest energy density of all fossil fuels and hence it is a truly global resource, with production ranging from the Arctic coasts to equatorial forests and hot deserts, and with enormous investment in an unmatched worldwide shipping infrastructure (long-distance pipelines, oil loading and offloading terminals, giant tankers) and high-throughput processing in large refineries.

In contrast, wood and crop residues (mostly cereal straws), the two most common traditional phytomass fuels, have low energy densities, with crop residues at just 15 MJ/kg while wood (depending on species and the degree of dryness) ranges from less than 15 MJ/kg (for fresh-cut branches) to about 17 MJ/kg for air-dry wood and to almost 20 MJ/kg for absolutely dry woody matter. Charcoal is the great exception as the pyrolysis of wood produces nearly pure carbon with specific energy of nearly 30 MJ/kg. This means that twice as much straw or fuelwood has to be burned to yield the same amount of energy, and charcoal’s smokeless combustion (as opposed to often very smoky burning of wood in open fires or poorly designed stoves) was another welcome advantage for indoor use. But in traditional societies there was a high energy, and environmental, price to be paid for these advantages as charcoal-making in simple clay kilns required as many as ten kg of wood for every kg of charcoal.

Coal may be as energy dense as charcoal: its early extraction often produced the highest quality anthracites (much like charcoal a nearly pure carbon with specific energy of up to 30 MJ/kg) and excellent bituminous coal (25-27 MJ/kg). As coal mining progressed, average energy content of produced coal has declined, particularly with the shift to less expensive and much safer surface extraction. Specific energies of most steam coals now fit within the range of 22-26 MJ/kg, but those of the poorest lignites are less than 10 MJ/kg. Transition to liquid hydrocarbon introduced fuels of unrivalled energy density: crude oils range from gasoline-like light liquids to heavy varieties that might need heating for transportation but their specific energies span a narrow range of 42-46 MJ/kg; with densities ranging between 0.75-0.85 kg/m3 this translates to 32-39 MJ/m3. Because of these high densities refined liquid fuels dominate road and water transportation.

Natural gas (mostly or purely methane, CH4) has a higher hydrogen share (75%) than liquid hydrocarbons and hence it contains 53.6 MJ/kg –- but to liquefy it requires considerable energy input for refrigeration and this option is used only for (still expensive) intercontinental shipments of liquid natural gas (LNG). In its gaseous form methane’s energy density is 35 MJ/m3, amounting to less than 1/1,000 energy density of gasoline. Higher hydrogen content explains this progression of higher energy density. Very low energy density of natural gas is no problem when the fuel is delivered by pipelines for stationary combustion in electricity-generating gas turbines or in industrial, commercial and household furnaces.

Finally, in the third category of density are those derived SI rates that relate a basic quantity (current, luminous intensity) or a derived unit (to space: they include current density (A/m2), luminance density (cd/m2), illuminance density (lm/m2), electric flux density (C/m2) and magnetic flux density (Wb/m2). Power density is not an official name given to any derived quantity on the list of SI units. The rate –- measured in W/m2 -– is listed among the derived quantities but it is given a rather restricted scope: it is called heat flux density or irradiance (flux of solar or other radiation per unit area). But in this book power density always refers to the quotient of power and land area, and I will demonstrate that this rate is a key variable in energy analysis because it can be used to assess suitability and potential of specific energy resources, performance and operating modes of energy converters, and requirements and structures of complex energy systems.

In all of these respects energy density is an insufficiently revealing measure. For example, crude oils (regardless of their appearance and physical differences) have uniformly high energy density but if present in minuscule reservoir strewn over a large area power density of their extraction would be too low to warrant their commercial exploitation; in contrast, the Middle Eastern oil fields supply the whole world precisely because they produce high energy density fuel with unmatched power densities. Similarly, US bituminous coal, mined with high power densities, is shipped to Europe where energy-dense hard coal deposits remain unexploited because their thin seams could not be extracted with sufficiently high power density.



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, telecommunication 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, the concept that can be used to assess potential performance of all modern commercial energy conversions. And in this book I will 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 mass (W/g) terms, are used to rate the performance of batteries. While energy density (J/g) measures the specific energy a battery can hold (the higher rate implying, obviously, a longer runtime), power density measures the maximum energy flux that can be delivered on demand in short bursts of electricity required for tools, medical devices and in transportation. Early lead acid batteries delivered less than 50 W/kg, by the end of the 20th century power densities of these massively deployed (above all automotive) units were between 150-300 W/kg. Utility-grade batteries can deliver up to 40 MWh of electricity with efficiencies of up to 80% and their power densities range between 200-400 W/kg at 80% charge level. Since the year 2000 the best performance of experimental lead-acid batteries has been boosted by the addition of high-surface area carbon: its addition adds only up to 3% of weight but it increases surface area by more than 80%) to more than 500 W/kg and the ultimate target is 800 W/kg (Svenson 2011).

More expensive nickel-cadmium batteries can supply up to about 1 kW/kg and increasingly common lithium-ion batteries deliver well in excess of that (Rosenkranz, Köhler and Liska 2011; Omar et al. 2012). High-energy 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) deliver less than 80 Wh/kg but have power densities up to 2.4 kW/kg at 80% charge; as the density of these batteries is about 2.1 kg/L that translates to almost 5 kW/L. By 2013 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 average core power density is the amount of energy generated by 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 (that is kW/L) and the range for all reactors listed in the IAEA’s PRIUS 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 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 70-110 kW/dm3. The highest rates, in excess of 700 kW/dm3, have been achieved in molten salt-cooled experimental fast breeder reactors (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 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 the human exposure levels to non-ionizing radiation do not exceed accepted safety standards. The US Federal Communications Commission set the maximum permissible exposure (MPE) for power density for transmitters operating at frequencies between 300 kHz and 100 GHz (FCC 1996).

Between 30-300 MHz (very high frequency wavelengths of 1-10 m that carry FM radio and TV broadcasts) the limit is 1 mW/cm2 for occupational exposure averaging 6 minutes, and just 0.2 mW/cm2 for general population and uncontrolled exposure averaging 30 minutes. For short-wave 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 short-wave radio transmitters have power of 50-500 kW, while many long-wave transmitter rate more than 500 kW and the world’s most powerful one requires 2.5 MW. 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/4πr2. A 100-kW transmitter would thus produce PD of 0.8 nW/m2 at the distance of 1,000 km, equal to only one-millionth of the allowable exposure.

In reality, most radio antennas have considerable transmission gain (Gt) created by suppressing upward and downward directions and concentrating the output toward the horizontal plane. After correcting for this intervention (PD = PtGt/ 4πr2) a 100 kW short-wave transmitter with a gain factor of 10 will have effective radiated power of 1 MW and 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 high gain of up to 30 or 40 dB (that is Gt between 1,000 and 10,000) in order to pinpoint distant targets (Radartutorial 2013).

For ultra-high frequency used for cellphones (1.9 GHz) the general population limit is 1 mW/cm2 while the actual received maxima near a cellphone tower are 10 μW/cm2, that is just 0.01 mW/cm2. Higher power density exposures apply to time-varying electric, magnetic an electromagnetic fields between 30 and 300 GHz: these wavelengths of 1-10 mm are the highest radio frequency just below the infrared radiation. International Commission on Non-ionizing Radiation Protection puts the maximum power densities at 50 W/m2 for occupational 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 asses power density of virtually all energy conversions by quantifying energy flux per unit area of the converter’s surface. In 1884 John Henry Poynting (1852-1914), 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 π 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) formulated the same concept a decade earlier (Umov 1874) and hence in Russia the measure has been known as Umov-Poynting vector. Piotr Leonidovich Kapitsa (1894-1984) -– Russian Nobel Prize winner in physics (in 1978, for his basic discoveries in the area of low-temperature physics) and Ernest Rutherford student –- pointed out that the vector can be used to assess all energy conversions in order to reveal “particular restrictions of these various flows’’ that are often ignored, resulting ‘’in wasting money on projects that can promise nothing in the future’’ (Kapitsa 1976, 10).

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 wind speed v at 12 m/s and air density (at 200 C) of 1.2 kg/m3 kinetic energy density (U = 0.5mv3) will be 1,037 J/m3 and the maximum power of the machine –- would be 9.14 MW.


Maximum power of a wind turbine
P = ½ρAv3

= 0.5 x 1.2 x 7,854 x (12)3

= 4712.4 x 1728

= 9.14 MW


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 due to unavoidable energy losses (in gearing, bearings) and the correction factors range between 0.35-0.45 even for the best designed modern wind turbines. I chose the blade radius of 50 m because it matches that of GE 2.5 MW series turbines (GE Power & Water 2010): their guaranteed power means that the actual power coefficient of these large machines is only 0.3, and that the effective power density of their electricity generation is 318.3 W per 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, famous physicist Abram Fedorovich Ioffe, 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 feeding to the electricity grid. But in order to avoid sparking the electrostatic field is restricted by air’s dielectric strength and to generate 100 MW (a rate sufficient so supply electricity needs of nearly 80,000 average American consumers) the electrostatic rotor would have to have area of about 400,000 m2 (nearly 0.5 km2), obviously an impossible requirement.

Maximum power than 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 (rate easily needed by a large city) would have to be on the order of 100,000 m2. Obviously, power density of fuel cells is too low to provide centralized base-load supply in 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) high velocities and temperatures of the working medium (steam superheated to 600 0C, travelling at 100 m/s with 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 will return to Umov-Poynting vector densities when reviewing 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 new Encyclopedia of Energy (Cleveland et al. 2004). Four authors use the measure in four different ways: in the first volume 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 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 2004, 613), as ‘’average long-term power flux per unit of land area, usually expressed in watts per squared meter’’; and in the sixth volume Grübler (2004, 163), writing about transitions in energy use, defines it as “amount of energy harnessed, transformed or used per unit area.”




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