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I have a fictional civilization that lives on a continent (or world) with a perpetually cold, wintry climate. The continent does have a large number of volcanoes of varying sizes and levels of activity.

This civilization builds their cities near subterranean magma chambers close to the planet's surface (which volcanoes would be a sign of) as a source of energy. Their drilling technology is not particularly sophisticated, so they can only really exploit the heat from shallow chambers. All their power needs—electricity and heat—are met geothermally.

So, my question is this: what volume would a shallow subterranean magma chamber (its roof 1000 meters below the surface) need to be in order to provide geothermal power for a city of 100k people who each use around 40,000 kWh of power a year?

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    $\begingroup$ Do you mean "big" by volume? Or by dimensions of the magma chamber? $\endgroup$
    – Alastor
    Commented Feb 2, 2023 at 4:18
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    $\begingroup$ They do not want to drill into a magma chamber. Magma tends to flow back up the hole and gum up everything with solid rock. Even in Iceland, they only want to get the hole near the magma so that enough heat is available. Unfortunately, they actually struck magma. bbc.com/future/article/… $\endgroup$
    – David R
    Commented Feb 2, 2023 at 15:36
  • $\begingroup$ @ArktourosUltorMaximus7600 I'm fine with either, but I think volume is probably good enough for my purposes. I'll edit my question. $\endgroup$
    – AustinZ
    Commented Feb 3, 2023 at 5:32

2 Answers 2

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To estimate the required volume of a shallow subterranean magma chamber to provide geothermal power for a city of 100,000 people, we need to consider several factors, including the average energy demand, the thermal output of the magma, and the efficiency of the geothermal power plant.

Assuming each person uses 40,000 kWh of energy per year, the total energy demand for the city would be 4 million MWh (100,000 x 40,000). To convert this to thermal energy, we would need to multiply it by the conversion factor, which is typically around 0.3 (30%) for a geothermal power plant. So, the required thermal energy would be 1.2 million MWh/year (4 million x 0.3).

Next, we need to consider the thermal output of the magma. The heat flow from a magma chamber depends on its temperature, the area of the chamber, and the heat transfer coefficient. The temperature of the magma can range from 600°C to over 1000°C, and the heat transfer coefficient can range from 0.1 to 0.5 W/m². Based on these assumptions, a rough estimate of the thermal output of the magma chamber can be calculated.

Finally, we need to factor in the efficiency of the geothermal power plant, which can range from 20% to 40%. Based on this, we can calculate the required volume of the magma chamber.

Let's use some rough assumptions based on typical values for the temperature of the magma (700°C), heat transfer coefficient (0.2 W/m²), and the efficiency of the geothermal power plant (30%).

The thermal output of the magma chamber can be calculated as follows:

Q = kA(Tm - T)

where Q is the thermal output, k is the heat transfer coefficient, A is the area of the magma chamber, Tm is the temperature of the magma, and T is the temperature of the surrounding rock.

Assuming an average area of 1000 m² for the magma chamber, the thermal output can be estimated as:

Q = 0.2 x 1000 x (700 - 20)

Q = 112,000 W

To convert this to thermal energy, we multiply it by the number of hours in a year:

E = Q x 24 x 365

E = 3.2 million MWh/year

To meet the energy demand of the city, we divide the total thermal energy required (1.2 million MWh) by the thermal output of the magma chamber (3.2 million MWh/year):

V = 1.2 / (3.2 / 1000)

V = 375 m³

So, the volume of the magma chamber would need to be approximately 375 m³ to meet the energy demand of the city, based on these assumptions.

It's important to note that these calculations are rough estimates based on assumptions and should not be taken as exact values. The actual volume required could vary significantly based on various factors such as the efficiency of the geothermal power plant, the heat flow from the magma chamber, and the temperature of the magma.

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  • $\begingroup$ Thank you so much for this! I appreciate the thorough explanation. $\endgroup$
    – AustinZ
    Commented Feb 13, 2023 at 5:38
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This entirely depends on the technology used to extract heat, but you don't generally need a magma chamber at all to extract geothermal energy.

The technique I've seen:

  1. Bore a hole about a mile down.
  2. Lower a big bomb (sometimes thermonuclear) to the bottom of the hole and detonate it. This generates a large volume of rock that is solid, but warm enough to make high pressure steam.
  3. Cap off the bore hole with power generation equipment.
  4. Drill a hole into the fracture zone sideways.
  5. Pour water down the second hole.

The second hole is cooler, so it has greater water pressure than the first one. Steam comes out of the bore hole, and you generate power off of it just like you would a locomotive boiler.

There are numerous technical issues that we still need to solve. The one cited in the article I read was that they never figured out how to seal the heating chamber, so they lost too much water. Another is that you have to find an area with the right temperature. Too deep, and the pressure has to be too high, and the outlet becomes unpredictable. Too shallow and the weight of the water in the second hole isn't enough to keep the water from backflowing.

I'm sure your people could figure it out.

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