Heat and the Silo
I was interested in the thermodynamics and power requirements for the hive, so I started doing some quick estimates of what it would take to heat, cool, and power a Silo. There are a lot of assumptions here, but if you're the sort of person who likes numbers, it's a fun afternoon's exercise. The goal here isn't to say 'this is how it must be done' but to present a plausible mechanism to solve the problem.
[If you're more of a 'it's not that kinda show' sort, that's fine. This is probably not the post for you. 😄 ]
My main assumption is that the Silo is a quasi-closed system. We know that liquid water comes in (groundwater infiltration), steam comes in (for the turbine-generator), and water gets pumped out. Every bit of energy added to the Silo eventually becomes heat, and the only way out for that heat is conducting out via the walls or being pumped out.
My main findings:
- Food production is, by far, the dominant consumer of energy and main producer of waste heat
- The known rate of water infiltration is not enough to cool the silo at ambient temperatures (e.g. inlet ~22C outlet ~30-35C). To get within an order of magnitude of heat balance, you would need heat pumps to boost the outlet temperature to be above the boiling point of water, allowing a phase change and a much higher heat absorption capacity.
- But, it's still not enough. To achieve balance, agriculture would need to be much more efficient on a molecular level. Luckily, the silo was built by a man who makes unicorns for fun, so this might be plausible within the story.
Here are the more detailed estimates for the baseline silo (normal crops). These are the average constant heat production numbers:
| System | Thermal Dissipation |
|---|---|
| Food Production & Processing | 268.9 MW |
| Industrial, Commercial, Residential Power | 7.8 MW |
| Air Handling | 3.6 MW |
| Water Handling | 1.3 MW |
| Amenities, other | 3.6 MW |
| Conductive heat flux | 0.8MW |
| Electricity Thermal Waste | 58.9 MW |
| Total | 345.1 MW |
Air handling includes ventilation, intra-silo cooling, dehumidification, and odor removal. Water handling includes infiltration pumping, water purification, 'local' plumbing, sewage and wastewater treatment. Amenities includes things like IT, lighting, and a 2MW 'misc.' allotment. Conductive heat flux is the heat that is conducted into the silo from the warm rocks deep in the ground (or out for the cooler rocks near the surface). Electricity thermal waste is mainly generation losses outside of the steam path (e.g. gearbox, bearings, turbine casing, etc...) and also in distribution (wire losses, transformer losses, etc...).
Food is by far the dominant factor. The basic analysis is, each person needs about 3500 Calories (or kcal, about 14.6 Megajoules) per day. With high-efficiency LED grow lamps and a selection of efficient, nutritionally complete plants, an average electricity-to-edible-calories ratio of about 1.5% is reasonable. We also know they have some food animals (cows, rabbits, chickens, etc...) which can provide meat, eggs, milk, and Sims' leather jacket. But, you have to feed animals, and the feed-calories-to-edible-animal-calories adds more inefficiency. And there are some crops like fruit orchards that are probably less efficient, but help bolster morale (and are a source of wood and provide juice for Anthony Sims). And, mushrooms can be used and are very 'power' efficient. Making some assumptions about those (12% of calories from animal sources, 15% from 'luxury' crops, 4% mushrooms) and its about 25.3MJ per person per day (+ a bit for mushrooms). At 1.5% electricity-to-calories efficiency, and we're at 1.68 GIGAjoules per person per day. We also need about 20% more to run condensers to collect transpiration water and I assumed another 15% for non-cooking processing, hydroponics pumping, waste, etc... and that comes to about 26.9 kiloWatts per person, or 269 Megawatts for the whole silo. For comparison, that's about the residential power requirements for a US city of 214,000 households or about 540,000 people - ~50x more than the Silo's 10,000.
Figuring out how much water flows into the Silo would require a detailed geological survey and numerical simulations, but we can estimate something reasonable from Silo 17. Assume the silo has a radius of 75m (about 250ft), and is 85% empty space *Note1*. The Silo 17 rebellion was ~30 years ago. 100 of 144 levels have flooded. Putting that together (and adding some extra volume for the mines and digger void), we get an average of about 21kg of water per second since the pumps were blown. But, the water infiltration rate should be slowing down as the existing water creates backpressure. Based on Darcy's Law, I'd estimate the 'normal' (unflooded) infiltration rate to be about 40kg per second. But, that also wouldn't work. It is not clear if it rains anymore, and with all those silos clumped together, it would quickly deplete the local ground water *Note2*. To maintain a steady waterflow, water pumped out of the silo would have to be moved somewhere, cooled and then injected back near the silo at near Atlanta ambient temperatures. When the pumps stop (as happened in Silo 17) the ground water partially depletes and within a few years it reverts back to steady 'natural' state. But, when the pumps are running, the infiltration would be about 55% higher, or about 63kg per sec. [Note: this is the weakest part of the analysis since there are a lot of assumptions piled on each other, but I think these numbers are plausible.]
Atlanta is in the Piedmont geographic region. The first ~300m of subsurface are more fractured and have more water permeable rocks and soils. The 300-1600m region is less permeable and made from hard stones which are good insulators. I assume that most of the groundwater is coming from the top 300m which is cooler, but a substantial part comes from the lower, warmer regions. Averaging across, I assumed the average infiltration water is about 22C.
If we heat the incoming water from 22C to 35C, at 63kg per sec we can carry away about 3.4MW of heat. That's about 1% of the energy we need to remove from the silo.
Since the water's heat capacity is our limiting factor, the best thing to do is boost it. There have been several studies proposing to use heat pumps to boost the output temperature of cooling systems, mainly for lunar space colonies. It's a tradeoff - a heat pump increases the output temperature, so you can make your radiator smaller, but it adds complexity and you have to use more electricity to run the heat pump, which means you have to remove that from the system as well. The higher the temperature 'lift' the less efficient the heat pump gets.
But, if we boost the output temperature of the water to 105C steam, we get a phase change (i.e. boiling) which means it can carry a lot more heat. Now our 63kg per s can carry about 163MW of power.
But... we need another 221MW of power to run the heat pumps. So now we have our original 345MW + 221MW of heat pump power and only ~163MW of water-steam heat capacity. Fail.
To fix this ~3x shortfall requires some sci-fi elements. Per Stensen is a nano-tech and genetics expert who can hire a lot of good engineers and scientists. I think we can make some plausible assumptions about what he might do to improve things.
The biggest change has to be to agriculture - it consumes by far the most power. The figure of merit here is the electricity-to-edible-calories efficiency. Currently, 1.5% is reasonable for well selected crops. But, if you can tinker with plant DNA and nanoelectronics enough, there are some things that could improve that. Developing slightly more efficient deep-red LEDs, decreasing the amount of blue light required, Far-red photosynthesis via cyanobacteria cross breeding, better carbon fixing, better light harvesting, hydroponic adaptations like smaller stems and roots, and larger fruits, seeds, and tubers might sort-of feasibly (in a sci-fi sort of way) boost electricity-to-calories to about 7%. And, he can probably boost high-temperature heat pumps from 45 to 50% of the Carnot efficiency (heating COP).
This decreases the food production power (direct power and condensers), makes the heat pumps more efficient, helps air handling a bit (less ventilation power), and brings our total power:
| System | Thermal Dissipation (Old) | Thermal Dissipation (Improved) |
|---|---|---|
| Food Production & Processing | 268.9 MW | 58.1 MW |
| Industrial, Commercial, Residential Power | 7.8 MW | 7.8 MW |
| Air Handling | 3.6 MW | 3.1 MW |
| Water Handling | 1.3 MW | 1.3 MW |
| Amenities, other | 3.6 MW | 3.6 MW |
| Conductive heat flux | 0.8MW | 0.8 MW |
| Electricity Thermal Waste | 58.9 MW | 13.8 MW |
| Heat Pumps | 221 MW | 45.3 MW |
| Total | 556.1 MW (345.1 MW wo pumps) | 133.8 MW |
About 18% under the 163MW heat capacity of our outlet water, making it sorta-kinda feasible.
Some notes:
- There are probably additional savings to be had by using plants to process waste water, clever use of waste heat for cooking, using microbes or yeast for food, thermocompressors instead of electrical heat pumps, superheating the turbine exhaust, better management of transpiration water, etc... I think that adding some of those optimizations will improve the margin, but not change the main conclusions.
- I don't explicitly model cooking, but assume it fits somewhere in residential and commercial energy use. Probably more is cooked in cafeterias, which would be more efficient than residential cooking, which might save a little.
- Per could also probably make cows, chickens, and mushrooms a little more efficient too and boost some of the efficiency of local condensers and such, but I'm not adding that in.
- I am neglecting the waste heat from the turbine that is carried away in the exhaust steam (from e.g., isentropic inefficiency, blade friction, leakage, partial admission, throttling, disc windage, etc...). I assume this heat can be vented along with the heat pump steam. I don't think the turbine system was condensing, so the hot output steam from the turbine could combine with a 105C exit temp from the heat pumps and just convect upwards.
- I am neglecting stuff going on outside the silo (condensing and cooling water, generating power for IT, etc...) since they happen outside the silo and doesn't impact its thermodynamics.
- Because I'm assuming some reinjection of water into the ground in 'normal' operations, it could be argued that the Silo could just have a pipe directly in with cooled water. This would be a completely reasonable design, but because the text doesn't specifically mention a 'magic water cooling pipe' like the 'magic steam pipe', I thought it would be safer to assume indirect reinjection. The specific 55% boost comes from picking a reinjection site that is equidistant from neighboring silos (assuming the '200 foot' rule for the limitation on the mines)
- 3500 Calories per day is a lot compared to modern diets, but I'm trying to be conservative. Also, there are a lot more stairs to climb in the silo and probably a lot more manual work. 3,000 is probably a more defensible number, but I don't think it changes the overall conclusions.
- To decrease the temperature 'lift' of the high-temp heat-pumps, the input can be boosted by direct cooling of 'hot' things (e.g. hot side of condensers, turbine gearbox, etc...). So rather than lifting from the Silo's ambient 20-30C internal temperature, the average input is closer to 38-41C.
- There are going to be issues with scale removal on all these heat pumps and boilers. A more clever and complete design might have some blowdown mechanism built in. Maybe the scale is collected and pumped out in a high-speed jet of water? This might also be a good way get rid of the mine spoil and tailings. Either way, its beyond the scope of this analysis.
- Mushrooms: The 4% of calories from mushrooms is assuming it is limited by the amount of waste inedible organic residue (e.g. stems, leaves) from the staple crops. Because some of the genetic modifications reduce these things to make plants more 'hydropoics friendly' and efficient, the final case has only about 2% of calories from mushrooms. If that were not a limit, people can consume 10-15% of their calories from mushrooms without digestive difficulties.
*Note 1*: Width of the silo. It is not specified and various clues seem to be inconsistent. The Relic Hard Drive makes it look like a 75m radius, the blue prints Troy was looking at seem like ~110m radius, others have suggested 160m. I went with 75m to be conservative.
*Note 2*: I'm not a hydrologist, but based on a simple deep pit water infiltration estimate I found, the 'radius of influence' (distance at which a well impacts the level of groundwater) would be almost 900m, so Silos would have to be almost 2km apart to avoid depleting each other's groundwater. They don't appear to be that far apart.