Warm and Fuzzy On Geothermal?

Editor’s Note: EarthTechling is proud to repost this article courtesy of Do The Math. Author credit goes to Tom Murphy, a physics professor in San Diego, California.

The Earth started its existence as a red-hot rock, and has been cooling ever since. It’s still quite toasty in the core, and will remain so for billions of years, yet. Cooling implies a flow of heat, and where heat flows, the possibility exists of capturing useful energy. Geysers and volcanoes are obvious manifestations of geothermal energy, but what role can it play toward satisfying our current global demand? Following the recent theme of Do the Math, we will put geothermal in one of three boxes labeled abundant, potent, or niche (puny). Have any guesses?

The Physics of Heat

Thermal energy is surprisingly hefty. Consider that putting a room-temperature rock into boiling water transfers to it an equivalent amount of energy as would hurling it to a super-sonic speed! We characterize the amount of heat an object can hold by its specific heat capacity, in Joules per kilogram per degree Celsius (or Kelvin, since one degree of change is the same in either system). Tying some energy concepts together, the definition of a kilocalorie (4184 J) is the amount of energy it takes to raise 1 kg (1 liter) of water 1°C. So we can read the specific heat capacity straight away as 4184 J/kg/K. This is a rather large heat capacity, on the scale of things. As a rule of thumb, 1000 J/kg/K is a marvelously convenient universal number for most substances: it works for wood, air, rock, etc. Liquids tend to be higher (typ. 2000 J/kg/K), and metals tend to be lower (150–500 J/kg/K). Rocks—relevant for geothermal energy—range from about 700–1100 J/kg/K, and although I would be happy enough to use the convenient 1000 J/kg/K for crude analysis, I will be somewhat more refined and use 900 J/kg/K for rock in this post—although I feel silly for it.

image via Shutterstock

As an example, to heat a 30 kg dining room table by 20°C, we need to supply 600,000 J. Just multiply specific heat capacity by the mass and by the temperature change. A 1000 W space heater could do it in ten minutes (600 seconds), if all of its energy could be channeled directly into the table.

The next property to understand is thermal conductivity: how readily heat is transported by a substance. Differing thermal conductivity is why different materials at the same temperature feel like different temperatures to our touch. It’s because high thermal conductivity materials (metals) slurp heat out of our hands much faster than plastic or wooden objects would. Copper has a thermal conductivity of 400 W/m/K, while stainless steel has an abysmally low value (for a metal) around 15 W/m/K—which is one reason why stainless steel is the preferred metal in kitchens: we can tolerate holding the spoon or pot handle even when another part of the item is quite hot. Plastics are around 0.2 W/m/K, and foam insulation tends to be around 0.02 W/m/K. Rock falls between 1.5–7 W/m/K, with 2.5 W/m/K being typical.

How do we apply thermal conductivity? Imagine a flat panel of stuff with area, A, and thickness, t. Using the Greek letter kappa (κ) to represent thermal conductivity, the rate at which thermal energy flows across the panel given a temperature difference ΔT across it isκAΔT/t, which comes out in Watts.

Sources of Heat

Two sources contribute to the Earth’s heat. The first, contributing 20% of the total, derives from gravity. As proto-planetary chunks fell together under the influence of gravity, the kinetic energy they carried (converted gravitational potential energy) ended up heating the clumps that stuck together. If this were the only contributor, Earth’s center would have cooled significantly below its present levels by today. The other 80% of heating is the gift that keeps on giving: long-lived radioactive nuclei given to us by ancient supernovae (as with most of the other elements comprising Earth and ourselves). Specifically—in order of significance to heating—we have 232Th, 238U, 40K, and 235U, with half-lives of 14, 4.5, 1.25, and 0.7 billion years, respectively. Ironically, one can view the radioactive contribution as gravitational in origin also! This is because supernovae result from fusion losing the fight to gravity, and the heavy elements are created in the resulting gravitational collapse.

In total, the radioactive decay produces about 7×10−12 W/kg; in the mantle. The mantle occupies 85% of the volume of the Earth at an average density about 5 times that of water, having a mass of about 4.5×1024 kg. Multiply these together to get 34 TW of heat flow in steady state. If radioactivity is 80% of the story, this implies 42 TW total. Averaging over the area of the Earth, we get 0.08 W/m². Because of the decaying nature of radioactive materials, the heat generation was much higher a few billion years back, making Earth a more geologically active place (e.g., more volcanoes).

We can work up another estimate of the total geothermal heat flow by observing that the temperature gradient in the crust is 22°C/km. This gradient can be used as the ΔT/t part of the thermal conduction heat flow rate, κAΔT/t. Taking a square meter for A and 2.5 W/m/K for κ, we calculate a geothermal “loading” of 0.055 W/m². Indeed, Wikipedia reports a land-based heat flow of 0.065 W/m² while the ocean (due to thinner crust and thermally greedy water) averages 0.1 W/m².

Compared to Human Use

Using the Wikipedia value of 0.065 W/m² over land, multiplying by land area yields 9 TW. Humans use 13 TW currently. So if we managed to catch every scrap of land-based geothermal flow (and could use it efficiently), we would not fully cover our present demand. Needless to say, we’re not remotely capable of doing this.

1 Comment

  • Reply April 11, 2012

    Critical Thinker

    I dont know where you get all your facts but HP systems using geo-exchangers in the ground work very well….to say it doesnt work and you have to drill new holes,etc?….buddy, you need to be more educated on the technology, installation, and design before you make such statements….are a certified, accredited engineer on this subject?….around 50% of electrical use is towards heat/cooling….and this is not part of the solution?…lol….

Leave a Reply