The Azimuth Project
Power density (Rev #1)

The term power density is used in various ways, but in the work of Vaclav Smil it means the number of usable watts that can be produced per square meter of land (or water) by a given technology:

• Vaclav Smil, Power density primer: understanding the spatial dimension of the unfolding transition to renewable electricity generation.

This paper makes the point that switching to renewable forms of energy will require us to adapt to vastly lower power densities—a point also made by Saul Griffith in his discussion of ‘Renewistan’.

Smil writes:

Energy density is easy – power density is confusing.

One look at energy densities of common fuels is enough to understand while we prefer coal over wood and oil over coal: air-dried 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 a 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.

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. The world-wide web 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.

Here are some of Vaclav Smil’s results:

  • Most large modern coal-fired power plants generate electricity with power densities ranging from 100 to 1,000 W/m2, including the area of the mine, the power plant, etcetera.

  • No other mode of large-scale electricity generation occupies as little space as gas turbines: besides their compactness they do not need fly ash disposal or flue gas desulfurization. Mobile gas turbines generate electricity with power densities higher than 15,000 W/m2 and large (>100 MW) stationary set-ups can easily deliver 4,000-5,000 W/m2. (What about the mining?)

  • The energy density of dry wood (18-21 GJ/ton) is close to that of sub-bituminous coal. But if we were to supply a significant share of a nation’s electricity from wood we would have to establish extensive tree plantations. We could not expect harvests surpassing 20 tons/hectare, with 10 tons/hectare being more typical. Harvesting all above-ground tree mass and feeding it into chippers would allow for 95% recovery of the total field production, but even if the fuel’s average energy density were 19 GJ/ton, the plantation would yield no more than 190 GJ/hectare, resulting in harvest power density of 0.6 W/m2.

  • Photovoltaic panels are fixed in an optimal tilted south-facing position and hence receive more radiation than a unit of horizontal surface, but the average power densities of solar parks are low. Additional land is needed for spacing the panels for servicing, access roads, inverter and transformer facilities and service structures — and only 85% of a panel’s DC rating is transmitted from the park to the grid as AC power. All told, they deliver 4-9 W/m2.

  • Concentrating solar power (CSP) projects use tracking parabolic mirrors in order to reflect and concentrate solar radiation on a central receiver placed in a high tower, for the purposes of powering a steam engine. All facilities included, these deliver at most 10 W/m2.

  • Wind turbines have fairly high power densities when the rate measures the flux of wind’s kinetic energy moving through the working surface: the area swept by blades. This power density is commonly above 400 W/m2 – but power density expressed as electricity generated per land area is much less! At best we can expect a peak power of 6.6 W/m2 and even a relatively high average capacity factor of 30% would bring that down to only about 2 W/m2.