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Power Density Primer: Understanding the Spatial Dimension of the Unfolding Transition to Renewable Electricity Generation (Part I – Definitions)

[Editor’s note: This is Part I of a five-part series by Vaclav Smil that provides an essential basis for the understanding of energy transitions and use. Dr. Smil is widely considered to be one of the world's leading energy experts. His views deserve careful study and understanding as a basis for today's contentious energy policy debates. Good intentions or simply desired ends must square with energy reality, the basis of Smil's worldview.]

Energy transitions – be they the shifts from dominant resources to new modes of supply (from wood coal, from coal to hydrocarbons, from direct use of fuels to electricity), diffusion of new prime movers (from steam engines to steam turbines or to diesel engines), or new final energy converters (from incandescent to fluorescent lights) – are inherently protracted affairs that unfold across decades or generations.

Many factors combine to determine their technical difficulty, their cost and their environmental impacts. A great deal of attention has been recently paid to the pace of technical innovation needed for the shift from the world dominated by fossil fuel combustion to the one relying increasingly on renewable energy conversions, to the likely costs and investment needs of this transitions, and to its environmental benefits, particularly in terms of reduced CO2 emissions.

Inexplicably, much less attention has been given to a key component of this grand transition, to the spatial dimension of replacing the burning of fossil fuels by the combustion of biofuels and by direct generation of electricity using water, wind, and solar power. Perhaps the best way to understand the spatial consequences of the unfolding energy transition is to present a series of realistic power density calculations for different modes of electricity generation in order to make revealing comparisons of resources and conversion techniques. Detailed calculations will make it easy to replicate them or to change the assumptions and examine (within realistic constraints) many alternative outcomes.

Sorting Out the Definitions

Energy density is easy – power density is confusing. Energy density is simply the amount of energy per unit weight (gravimetric energy density) or per unit volume (volumetric energy density). With energy expressed (in proper scientific terms) in joules or less correctly in calories (and in the US, the only modern state that insists on using outdated non-metric measures, in BTUs), with weight in grams (and their multiples), and with volume in cubic centimeters, liters (dm3) or cubic meters, energy density is simply joules per gram (J/g) or joules per cubic centimeter (J/cm3) or, more commonly, megajoules per kilogram (MJ/kg) and megajoules per liter (MJ/L) or gigajoules per ton (GJ/t) and gigajoules per cubic meter (GJ/m3).

One look at energy densities of common fuels is enough to understand while we prefer coal over wood and oil over coal: air-dry wood is, at best, 17 MJ/kg, good-quality bituminous coal is 22-25 MJ/kg, and refined oil products are around 42 MJ/kg. And a comparison of volumetric energy densities makes it clear why shipping non-compressed, non-liquefied natural gas would never work while shipping crude oil is cheap: natural gas rates around 35 MJ/m3, crude oil has around 35 GJ/m3 and hence its volumetric energy density is thousand times (three orders of magnitude) higher. An obvious consequence: without liquefied (or at least compressed) natural gas there can be no intercontinental shipments of that clean fuel.

You can start explaining some of the limits and possibilities of everyday life or historical progress by playing with energy densities: the more concentrated sources of energy give you many great advantages in terms of their extraction, portability, transportation and storage costs, and conversion options. If you want to pack the minimum volume of food for a mountain hike you take a granola bar (17 J/g) not carrots (1.7 J/g). And if you want to fly across the Atlantic you will not power gas turbines with hydrogen: the gas has gravimetric density greater than any other fuel (143 MJ/kg) but its volumetric density is a mere 0.01 MJ/L while that of jet fuel (kerosene) is 33 MJ/L, 3,300 times higher.

Power density is a much more complicated variable. Engineers have used power densities as revealing measures of performance for decades – but several specialties have defined them in their own particular ways. The first relatively common use of the ratio is by radio engineers to express power densities of isotropic antennas as a quotient of the transmitted power and the surface area of a sphere at a given distance (W/m2). The second one refers to volumetric or gravimetric density of energy converters: when evaluating batteries (whose mass and volume we usually try to minimize) power density refers to the rate of energy release per unit of battery volume or weight (typically W/dm3 or W/kg); similarly, in nuclear engineering power density is the rate of energy release per unit volume of a reactor core. WWW offers a perfect illustration of this engineering usage: top searches for “power density” turn up calculators for isotropic antennas (the first common engineering use I noted above), and a Wikipedia stub refers to power density of heat engines in kW/L (the second common use as volumetric power density of energy converters).

To make it even more confusing, the international system of scientific units calls W/m2 heat flux density or irradiance, the latter referring clearly to incoming radiation (electromagnetic energy incident on the surface) – and Piotr Leonidovich Kapitsa, one of the most influential physicists of the 20th century (Nobel in 1978), favored using W/m2 for the most fundamental evaluation of energy converters by calculating the flux of energy through their working surfaces. The original late 19th century application of this measure (Umov-Poynting vector) referred to the propagation of electromagnetic waves but the same principle applies to energy flux across a turbine or to diffusion rates in fuel cells. Power density has been used recently in this sense in order to calculate a flux across the (vertical) area swept by a wind turbine (more on this in the wind power density section).

For the past 25 years I have favored a different, and a much broader, measure of power density as perhaps the most universal measure of energy flux: W/m2 of horizontal area of land or water surface rather than per unit of the working surface of a converter. Perhaps the greatest advantage of this parameter is that it can be used to evaluate and to compare an enormous variety of energy fluxes ranging from natural flows and exploitation rates of all energy sources (be they fossil or renewable) to all forms of energy conversions (be it the burning of fossil fuels or water- or wind-driven electricity generation). That is why I chose power density as a key analytical variable to evaluate all important biospheric and anthropogenic energy flows in my first synthesis of general energetics in1991 and why I had recently revised and substantially expanded that coverage in my book Energy in Nature and Society: General Energetics of Complex Systems. The MIT Press (2008).

Not many people have been using this powerful and revealing measure frequently and appropriately, hardly a surprise given the generally abysmal understanding of fundamental energetics. But, finally, power density expressed as energy flux per unit of horizontal surface has been receiving more attention because of the growing interest in renewable energy resources and their commercial conversions to fuels and electricity. Invariably, power densities of these stocks and flows are considerably lower than power densities and uses of fossil fuels, those highly concentrated stores of ancient photosynthetic production – and these differences are a key factor in determining the potential contribution of renewable energies to the world’s future fuel and electricity supply.

In this brief primer I will illustrate these contrasts by quantifying power densities of six modes of electricity generation: I will either make assumptions that closely correspond to representative modes of current operations or I will introduce actual generating facilities as typical examples. In subsequent posts I will first calculate the most common range of power densities of coal- and wood-fired (a renewable) electricity generation followed by power densities for natural gas-fired gas turbine-driven process; this will be followed by power densities of three new renewable conversions: a thermal station burning plantation-grown wood, solar photovoltaic plants, concentrating solar plants, and large wind farms.


Vaclav Smil is a Distinguished Professor at the University of Manitoba and the author of 30 books. His website is at http://www.vaclavsmil.com. For more on power densities consult Energy in Nature and Society and Energy Transitions.


1 Ben of Houston { 05.10.10 at 2:17 pm }

Professor, while I appreciate the detailed approach, you may want to simplify your language. Engineers and others who can follow your lecture are more likely to already understand this concept and sheer magnitude of area required for renewables. We need to convince the lawyers, politicans, bankers, and other non-technical people of power about what you are saying. I don’t think these people could follow even this introduction.

2 Robert Bradley Jr. { 05.10.10 at 11:14 pm }


Robert Bryce’s new book, Power Hungry, reviewed at MasterResource and available on Amazon, is a good, readable introduction to energy density.

3 Mark Heslep { 06.27.11 at 9:40 pm }

” … And if you want to fly across the Atlantic you will not power gas turbines with hydrogen: the gas has gravimetric density greater than any other fuel (143 MJ/kg) but its volumetric density is a mere 0.01 MJ/L ”

Yes, though spacecraft manage the trick quite nicely on a regular basis. That is, they do so w/ liquid hydrogen at 10 MJ/L when the energy overhead for the cryogenic process is not relevant, but the unsurpassed energy gravimetric density of hydrogen is, and as it is for atmospheric aircraft.

4 Atanda Gbengar { 07.19.11 at 12:08 pm }

I want the Content of Power and energy

5 Tony Echoi { 04.19.13 at 1:14 pm }

Please can someone help me? I need a standard definition of power density~THANKS

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