Power density

Power density of wood burning

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Power density of wood burning
10 t/ha x 19 GJ = 190 GJ/ha

190 GJ/31.5 Ms = 6,032 W

6,032 W/10,000 m2 = 0.6 W/m2

In order to supply a 1 GWe wood-fired power plant operating with the capacity factor of 70% and conversion efficiency of 35% would require an annual harvest of about 330,000 ha of fast-growing tree plantation, an equivalent of a square nearly 58 x 58 km:

Land claimed by a tree plantation
1 GW X 0.7 = 700 MW

700 MW/0.35 = 2 GW

2 GW/0.6 W/m2 = 3.33 Gm2 (333,000 ha)

√3.33 Gm2 = 57,735 m

Total area needed by a wood-fired electricity-generating plant is quite negligible when compared to a large area of land claimed by phytomass production: even if the generating station and its associated structures (fuel storage, switchyard, office and maintenance buildings, access roads) were to make an excessive claim of 10 ha (a square with sides of 316 m) it would still be only 0.003% of the land required to grow trees. Obviously, very low power densities of wood-based electricity generation prevent it from becoming anything but a very minor contributor to the overall supply, with plants usually burning wood waste generating by wood-processing industries.

If only 10% of the US electricity generated in 2012 (that is 405 TWh or 1.46 EJ) had to be fueled by wood, then (with average 35% conversion efficiency) the country would require about 4.17 EJ (about 132 GW) of wood chips. With average power density of 0.6 W/m2 this would claim about 220,000 km2 of wood plantations, or nearly as much land as the entire Idaho or Utah. This calculation also shows why even assumptions of major, but realistic, yield increases (thanks to better hybrids or to entirely new transgenic trees) would not make a fundamental difference as far as the overall land claims are concerned the very low power densities of phytomass production are concerned. Even with the doubling of the assumed 10t/ha mean (not likely in on a large scale in temperate regions for decades to come) a wood-fueled plant with 1 GWe capacity would still need annual harvests of all above-ground phytomass grown in a plantation of nearly 170,000 ha, a square of 40 x 40 km.

Liquid biofuels Similar simple calculations reveal the power densities of electricity and fuels produced from agricultural crops. Some of these plants have seen impressive gains in yields during the second half of the 20th century. Before the introduction of hybrid varieties the US nationwide corn yield averaged just 1.5 t/ha during the mid-1930s, by 1975 it was 5.4 t/ha, in 2000 it reached 8.6 t/ha and it peaked at 10.3 t/ha in 2009 (FAO 2014). Similarly, better cultivars and better agronomic management of Brazilian sugar cane helped to lift its average yields from 43 t/ha in 1960 to 71 t/ha in 2012 (FAO 2014). By far the most process to convert these crops to fuel has been the production of ethanol and it, too, has seen improvements over time: for example, Brazil’s ethanol yield increased from 59.2 L/t of sugar cane in 1975 to 80.4 L/t in 2008 (Cortez 2011).

Experiments with automotive ethanol predate WW II but modern large-scale ethanol production began in Brazil with the country’s sugar cane-based PROALCOOL program in 1975, and in the US with corn-based effort in 1980 (Basso, Basso and Rocha 2011; Solomon, Barnes and Halvorsen 2007). By 2010 the two programs expanded to, respectively, to about 50 and 20 GL a year, but Brazilian production has been stagnating since 2008 (Angelo 2012) and the supply of the US corn-based ethanol is unlikely to grow beyond 9-10% share of automotive market it has reached thanks to the production mandates imposed by the US Congress because the future fate of these obligations is uncertain.

America’s ethanol producers have been putting the average conversion ratio at 2.8 gallons per bushel of wet corn (ACM 2013); in metric units this would be 0.38 kg/kg of dry grain but Patzek (2006) showed that the theoretical yield of ethanol from corn starch (making up 66.2% of the grain) is 0.364 kg/kg of dry corn. This discrepancy is explained by the industry’s practice of counting gasoline denaturant (5% by volume, 8% by energy content) as ethanol. Using the yield of 0.36 kg of ethanol (energy density of 26.7 MJ/kg) per kg of dry grain and assuming average US harvest of 10 t/ha results in power density of 0.26 W/m2 for the US corn-based ethanol.
Power density of US ethanol production
Grain corn yield

(15% moisture) 10 t/ha

Moisture adjustment 10 x 0.85 = 8.5 t/ha of dry grain

Ethanol yield 8,500 x 0.36 = 3,060 kg/ha

Energy yield 3,060 x 26.7 MJ/kg = 81.7 GJ/ha

Power density 81.7 GJ/ha = 2,590 W/10,000 = 0.26 W/m2

Sugar cane is a perennial grass but because its yields decline with successive cuttings the standard Brazilian practice is to have five harvests followed by replanting (SugarCane.org 2013). According to Crago et al. (2010) for the first harvest ethanol yield is 10,235 L/ha and it diminishes to 5,636 L/ha with the fifth harvest: the average is 6,134.4 L/ha, corresponding to power density of 0.41 W/m2, nearly twice as high as for the corn-based US ethanol. Sugar cane ethanol has other advantages: the land used to grow the grass is not in competition with land used for food crops, the plant’s endophytic nitrogen-fixing bacteria eliminate the need for nitrogen fertilizers, and in Brazil’s climate there is no need for irrigation. Production costs of sugar cane ethanol are also cheaper than those of the US corn ethanol but the relative competitiveness of the two fuels is changed by including transportation costs of the former and by-product (distiller’s grain, corn oil) credits of the latter (Crago et al. 2010).

The other kind of liquid biofuels that is now in commercial production is the conversion of oil extracted from seeds of oil plants into biodiesel. This is done by transesterificaction, that is by adding plant oil to heated mixture of a catalyst (KOH) to ethanol (or methanol) to produce ethyl or methyl ester after removing heavier glycerin (Gerpen 2005). Transesterification process converts up to 97% of oil into biodiesel, and rapeseeds contain about 40% oil which means that nearly 39% of the crop yield can end up as fuel. The crop yields mostly between 2-3.5 t/ha (EU-27 mean is about 2.5 t/ha) but the average is 4 t/ha in the Netherlands.

The Dutch biodiesel yield thus averages close to 1.5 t/ha and its energy equivalent (with 37.8 GJ/t) of 56.7 GJ/ha translates to 0.18 W/m2, the best rapeseed-based performance by a Spanish company was 0.22 W/m2 (González-García, García-Rey and Hospido 2013) but the average EU yield translates to only 0.12 W/m2. As in the case of the US corn-based ethanol, it is obvious that this low power density alone precludes that this fuel could ever supply a significant share of the EU’s large diesel demand: rapeseed required to cover the EU-27 diesel demand of roughly 260 GW would have to be planted on nearly 220 Mha while the EU-27 arable land adds up to only about 103 Mha (Eurostat 2012). In any case, given the high energy cost of farming and processing inputs about a quarter of the EU’s area could produce rapeseed biodiesel only with net energy loss (Firrisa 2011).

Biofuel with the greatest promise –- but with constantly deferred large-scale production –- is cellulosic ethanol that is to be made by enzymatic hydrolysis of any phytomass high in cellulose and hemicellulose, a large category of tissues that ranges from any woody material to cereal straws and to intensively cultivated high-yielding grasses. The latter group includes switchgrass (Panicum virginatum), a North American native tolerant of summer heat; reed canary grass (Phalaris arundinacea), up to 2 m-high creeping C3 species from temperate Eurasia and North America; giant reed (Arundo donax), a common Euroasian rhizomatous plant that can grow up to 8-9 m miscanthus (Miscanthus giganteus), originally an Asian C4 species whose rapid growth can reach 3-4 m (Singh 2013; 4F CROPS 2011). In experimental plots and in small field settings these grasses yield mostly between 10-20 t/ha and dry matter yields of up 50 t/ha were reported for reeds –- but there is no doubt that without adequate fertilization high production rates could not be maintained in large-scale commercial plantings.

Potential of cellulosic ethanol has been chronically overestimated: completion dates of large commercial plants have been slipping for years, production costs remain uncompetitive, and conversion of hemicellulose, which makes up 25-36% of grass tissues (Lee et al. 2007), remains a challenge. But no future technical improvements can change the fundamentals. Even when assuming very high-yields (15 t/ha of dry matter) and average conversion of 330 L of ethanol/t of grass (Schmer et al. 2008) power density of cellulosic ethanol would be about 0.4 W/m2, no higher than that of Brazilian ethanol made from sugar cane, while the NREL’s design of thermochemical pathway by indirect gasification and mixed alcohol synthesis assumes the yield of 8.2 GJ/t of dry feedstock (Dutta et al. 2011), and that would translate to just 0.26 W/m2 for production based on harvesting 10 t of grain corn stover.

Finally, power density of biogas generation, an energy conversion technique that has been pioneered in rural areas of China and India as a means of turning organic (animal and plant) waste into low-energy density gas for household use, but that is now used on large commercial scale to produce gas for subsequent electricity generation. Among the affluent countries Germany has by far the most extensive national biogas program: in 2011 it had 7,000 biogas plants with installed capacity of about 2.7 GWe, and their feedstock was about equally divided between energy crops and livestock excrements (FNR 2012). Because of their very high water content cattle and pig slurry generate only a small share of biogas, with most of it coming from the fermentation of phytomass. German data show an equivalent of about 100 m3 CH4/t of fresh corn silage (FNR 2012), and with 50 t/ha this would produce 5,000 m3 of methane, or 0.6 W/m2 and after conversion to electricity (producing about 18.5 MWh) the final power density would be slightly above 0.2 W/m2.

Geothermal energy

Sources of the Earth’s huge heat flux are yet to be accurately apportioned: they include basal cooling of the Earth’s primordially hot core and, above all, heat-producing isotopes of 235U, 238U, 232Th and 40K in the crust (Murthy, van Westrenen and Fei 2003). At the ocean bottom heat flux is as low as 38 mW/m2 through very old (at least 200 million years) floors to more than 250 mW/m2 for ocean floors younger than 4 million years; on land the youngest crust has average heat flow of 77 mW/m2, the oldest continental shields average less than 45 mW/m2, and the weighted global mean is 87 mW/m2 (Pollack, Hurter and Johnston 1993).

Typical terrestrial heat flows are an order of magnitude lower than typical flows produced by photosynthesis of natural ecosystems, obviously too low to be converted to useful energy at the Earth’s surface. Drilling gets to progressively higher temperatures: geothermal gradient (the rate of temperature increase with depth) is usually 25-300 C/km but the extremes range from less than 150C/km to more than 500C/km in volcanic and tectonically active regions, particularly along the plate conversion and subduction zones surrounding the Pacific Ocean (Smil 2008). In selected locations in these regions hot (>1000C) pressurized water can be tapped by drilling often no deeper than 2 km and used (directly as steam or indirectly in binary systems where another liquid is heated by hot water in an exchanger) to produce electricity in turbogenerators (Dickson and Fanelli 2004).

In vast continental regions geothermal surface and near-surface fluxes are too small and their temperature is too low for rewarding conversion to electricity, but in many locations these flows can supply significant volumes of hot water for household, commercial and industrial uses, and almost everywhere shallow wells can be used for heat pumps. Of course, high-temperature water can be recovered anywhere after drilling sufficiently deep into the Earth’s crust but in regions with normal geothermal gradient it would require wells of at least 7 km deep to temperate in excess of 2000C where injected water would contact dry hot rocks.

Such enhanced geothermal systems (EGS) would be the most intensive way of geothermal capture as a set of deep wells reaching hot rocks would be connected by circulating injected water that would be heated to more than 2000C and withdrawn for electricity generation (Tester et al. 2006). Commercial exploitation of this option is yet to come, and all existing geothermal plants use hot water withdrawn from wells that are relatively shallow (less than 1km at Italy’s Larderello and New Zealand’s Wairakei) or of a medium depth (2-2.5 km in Iceland and California); these wells are sometimes recharged not only with fresh surface water but also with treated waste water.

Geothermal electricity generation began at Italy’s Larderello field in 1902; New Zealand’s Wairakei came only in1958, followed by California’s Geysers in 1960 and Mexico’s Cerro Prieto in 1970. Most of these pioneering installations were gradually enlarged, and other countries began to develop their considerable geothermal potential. By the year 2010 worldwide geothermal electricity-generating capacity reached 10.7 GW with nearly 3.1 GW in the US, 1.9 GW in the Philippines, 1.2 GW in Indonesia, 958 MW in Mexico, 843 MW in Italy, 638 MW in New Zealand, and 575 MW in Iceland; annual generation was about 67 TWh, resulting in a capacity factor of just over 70% (Bertani 2010; IGA 2013).

Wells in geothermal fields could be spread over areas of 5-10 km2 but (much as in the case of hydrocarbon extraction) well sites will claim only about 2% of the field’s area, and this claim can be further minimized with multiple wells drilled directionally from a single pad. Many gathering pipelines transporting hot water and steam are mounted on high supports and do not preclude grazing (or even crop cultivation) and movement of wild animals, and the plants themselves (generator halls, cooling towers, auxiliary buildings, switchyards) are fairly compact: holding ponds for temporary water discharges during drilling and field stimulation may be the largest land claim component. But pipelines and roads may cause some deforestation and they obviously contribute to fragmentation of habitat or (in some locations) to increased erosion and higher likelihood of landslides.

Geothermal Energy Association puts the aggregate claims for 30-year operation of a typical geothermal plant at 404 m2/GWh, that is 283 W/m2 (GEA 2013). DiPippo (1991) lists the land requirements of a 110 MW geothermal flash pant (excluding wells) at nearly 14 ha, or almost 800 W/m2 in terms of installed capacity; for a smaller (20 MW) binary plant (also excluding wells) his rate was about 700 W/m2, and power density for a 56 MW flash plant including all other infrastructures (wells, pipes, roads) it was about 135 W/m2. Finally, McDonald et al. (2009) puts the highest power density (most compact projects) of geothermal power plants (based on American data) at 113 W/m2, the least compact at just 8 W/m2.

As with other kinds of electricity-generating stations, land claims of geothermal plants are readily assessed by accessing the highest resolution Google Earth images; in addition, we have detailed land use studies at some major geothermal fields. California’s Imperial Valley has some of the most compact facilities located amidst crop fields: 40 MW Heber binary plant (using hot water to heat another working medium, pressurized liquid with lower boiling point to run the turbine) claims 12.15 ha (power density of about 330 W/m2 in terms of installed capacity) and a neighboring 47 MW Heber double-flash plant (with hot water vaporized in sequence under low pressure) occupies just 9.5 ha and its power density is almost 500 W/m2 (Tester et al. 2006).

Koorey and Fernando (2010) examined New Zealand’s geothermal projects (14 plants with total installed capacity of 890 MW) and found the lowest power density (in terms of installed capacity) of about 70 W/m2 at Ngawha 2008, and the highest one at Poihipi at 670 W/m2; Wairakei –- the country’s largest installation with 165 MW, using flash steam with binary cycle --- rated 133 W/m2; and the national mean was nearly 200 W/m2. All of these rates are for directly affected areas, that is (usually fenced) land exclusively occupied by power house, other plant installations and pipe routes. Addition of affected area, arbitrarily defined by the authors as all land 100 m out from the development area, lowers Wairakei’s power density to just 40 W/m2 and the national mean to 45 W/m2 -– but this count exaggerates the actual impact as about 40% of the affected areas are used as pasture and 14% are forested, an excellent example of concurrent land use.

Iceland’s largest geothermal development, Hellisheidi combined power plant (303 MWe) and heating plant (133 MWt, to be expanded to 400 MWt), taps into Hengill volcanic system with wells up to 4 km deep (Gunnlaugsson 2012).

The generating plant is supplied by 21 wells, each one supporting 7.5 MWe, and four drill holes are on a platform of just 1200 m2. The output density of hot water is thus 25,000 W/m2 but considerable infrastructure is required to turn that flux into electricity. The entire project –- including access and service roads, hot water production and fresh water injection wells, hot water, water and steam pipes, steam separator stations, power houses, cooling towers, steam exhaust stacks, water tanks, discharges and a switchyard system, injection areas and connection to the power grid –- covers 820 ha, and with 2.3 TWh of annual generation its capacity factor is nearly 87%. When counting only actual electricity generation Hellisheidi’s power density is just 32 W/m2; adding the available 400 MWt of heat raises that to about 660 MW of useful energy (electricity and heat) and implies overall power density of about 80 W/m2.

My final example is the world’s largest concentration of geothermal plants in the Geysers area of the Mayacamas Mountains north of San Francisco where 22 plants with the total capacity of 1.61 GW tap hot water from the productive area of 7,690 ha (BLM 2010). This implies average power density of just 21 W/m2, but most of the land are undisturbed mountains forests and meadows. Calpine Corporation operates 15 of the Geysers projects with about 700 MW of installed capacity and its infrastructure includes nearly 240 km of steam and water injection pipelines and about 270 km of access roads (Calpine Corporation 2013). Even with generous average claims (3 ha/plant, 5 m ROW for lines and 10m ROW for roads) the overall claim comes to about 1,250 ha and power density rises to 55 W/m2, very similar to New Zealand’s Wairakei.

Even the lowest power densities, those between 20-50 W/m2, are of the same magnitude as those of Alpine hydro stations, and an order of magnitude higher than those of the best large wind farms. Of course, these comparisons ignore different qualitative aspects of these three kinds of claims. In addition, and as in the case of underground coal mining, there may be also surface impacts following continuous withdrawals of hot water and reservoir recharging. With outflows much larger than recharges surface subsidence takes place in many geothermal formations: in parts of Wairakei field it was as much 45 cm/year (Allis 1990). Subsidence and landslides should not be a problem with EGS operations but some of them may experience induced seismicity.

While geothermal energy will remain a globally marginal source of electricity –- its share was a mere 0.33% of the global supply in 2010 (Bertani 2010) -– it can provide, locally and regionally, significant shares of household, commercial and industrial heat. More than 60 countries are now harnessing this resource. The US capacity of direct heat uses had more than tripled between 2000 and 2010, from about 3.8 GWt to 12.6 GWt; other large decadal increases have been in the Netherlands (from just 308 MW to 1.41 GW), Sweden (from 377 MW to 4.46 GW) and China (from 2.3 GWt to 8.9 GWt), and the global capacity increased from about 15.1 GWt in the year 2000 to 50.6 GW in 2010 while the actual energy use more than doubled to 438 PJ, implying average utilization factor of about 28% (IGA 2013).

Household heat pumps –- with relatively small individual capacity (on the order of 10 kW of heat for a typical American house) but large aggregate numbers due to their increasing popularity in affluent countries (in the US their total has surpassed one million) –- are the single largest category of these direct uses (Lund et al. 2003; Navigant Consulting 2009). They are followed by hot water for bathing and space heating; the most important commercial use are heating of greenhouses and aquacultural ponds, and a few percent of harnessed geothermal energy are used in industrial enterprises (for process heating, drying, evaporation, distillation, salt extraction etc.).

Space requirements of these installations (wells, piping, sometime also water tanks and heat exchangers) for commercial uses are minimal, easily accommodated within the fenced facilities. A typical American house will need three holes (up to 150 m deep) at least 6 m apart and with access space this will be at least 100 m2 and a heat flux of 100 W/m2. Horizontal closed loops buried at depth of 1-2 m cover at least 250 m2 (square of 15.8 m, easy to accommodate in most US suburban house lots, but too large for older, densely spaced urban housing) and yield heat transfer power density of 40 W/m2.


Fossil fuels

These energy sources are enormously concentrated transformations of biomass and hence the power densities of their natural occurrences are unrivaled by any other form of terrestrial energy. But their extraction, gathering, preparation, transportation and processing require a variety of permanent and temporary infrastructures that necessarily dilute the power densities of their delivery for final conversion to heat, motion and electricity.

All of these resources are abundant and although they are obviously finite (for more than a century the rate of their consumption has been surpassing the rate of their formation by many orders of magnitude) there is no imminent global danger of running out of coal or hydrocarbons in a few generations. Inevitably, progressing extraction of fossil fuels led some nations to shift from poorer, and more expensively produced, domestic resources to cheaper imports originating in giant surface coal mines and giant oil and gas fields, often on different continents. Many coal fields (some of them centuries old) have been entirely abandoned as underground mining of relatively thin seams became highly uneconomical, and an increasing shares of hydrocarbons are produced from greater depths, in locations more remote from principal markets or in deeper offshore waters. All varieties of fossil fuels thus became more expensive, but also more widely available and globally traded on massive scales and their aggregate extraction still keeps increasing.

Genesis of fossil fuels explains often extraordinarily high densities with which these resources are stored in the uppermost layers of the Earth’s crust. Coal seams are lithified and compressed layers of ancient phytomass: as most of its original oxygen and hydrogen was driven away by long spans of pressure and heat processing generations of those ancient swamp forests were concentrated into layers of carbon; some have remained remarkably pure, others were later adulterated with incombustible rock, and the younger coals still contain plenty of moisture. Relatively large shares of carbon present in ancient phytomass are preserved in coals, up to 15% for lignites and at least 10% for bituminous coals (Dukes 2003). Actual recovery factors will depend on the mining techniques (more on them later in this section) but the global mean is around 10%. When inverted this share means that about 10 units of ancient plant carbon (no less than five and as many as 20) were required to yield one unit of carbon in extracted coal.

Genesis of hydrocarbons has been entirely different: mixtures of liquid compounds accompanied by lighter gases originated through bacteriogenesis (anaerobic microbial metabolism) and thermogenesis (heat decomposition) of non-hydrocarbon organic molecules (carbohydrates dominated by cellulose, proteins and lipids) produced during the eras of high photosynthetic activity and buried in (marine or lacustrine) sediments; combination of these kerogen-forming processes can be dated in some shales as far back as 3.2 billion years ago (Rasmussen et al. 2008). Sedimentary source rocks may contain (by weight) as much as 10% of kerogen but 1-2% share is more common. Their prolonged heat decomposition (akin to producing lighter fuels in refineries) produced liquid hydrocarbons most of which were subsequently degraded by slowly acting thermophilic bacteria that convert lighter fraction to denser oils.

Formation of hydrocarbons preserves much lower shares of the original organic carbon than the genesis of coal, often less than 1% and only exceptionally more than 10% during the formation of oil-bearing sediments; after bacterial and heat processing this drops by at least another order of magnitude, and usually only a small fraction of mobile hydrocarbons migrates from the source rock to oil and gas reservoirs from which anywhere between 25-50% of fuels in place could be recovered. Using, again, inverted rates this means that on the order of 10,000 units of carbon present in the original biomass are required for every unit of carbon in extracted crude oil and 12,500 units are needed for every unit of carbon in produced natural gas (Dukes 2003).

Some natural gases require minimum processing before combustion (stripping H2O and H2S), but oils must be refined in order to produce the most desirable transportation liquids, gasoline, kerosene and diesel fuel. Inevitably, this further lowers the overall carbon recovery factor. Dukes (2003) summed up this low rate of carbon transfer by noting that every liter of gasoline (that is about 640 g C) requires about 25 t of initially sequestered marine biomass or at least 12 t C, corresponding to the transfer rate of a mere 0.005% and the requirement of roughly 20,000 units of initial biomass carbon to produce a unit of carbon in gasoline.

But given long periods of ancient biomass accumulations even these low transfer rates translate into often enormous energy storage in coals seams and in naturally pressurized oil and gas reservoirs, and huge amounts of oil remain bound in their source rocks, mostly in sands and shales. This means that even though some forms of fossil fuel exploitation are relatively space-intensive –- notably surface mining of coal buried under thick layers of overburden and pumping of oil from large numbers of low-productivity oil wells that require many access roads and gathering pipelines -– typical power densities of coal and hydrocarbon extraction are high and the enterprises producing these fuels claim relatively small amounts of land.

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