I thought I’d written about this somewhere, but I can’t find it, so I’m writing it again.
Adequate personal rapid transit would give low-density cities most of the benefits of high-density cities. In particular, what I have in mind are smooth 1.5-meter-diameter dirt-floored tunnels every 50 meters with pods — really electrified multi-person hybrids of horizontal elevators and bicycles — traveling through them at up to 28 m/s (100 kph or 60 mph in archaic units) under the control of a centralized just-in-time route planning system. Unlike a typical subway system, the pods move out of the track to park at a station; there are stations every 50 meters along each tunnel.
The pods are powered by overhead low-voltage DC power cables — 40 volts, for example, so it’s almost impossible for a human to receive a dangerous shock. At 28 amps, 40 volts supplies 1120 watts (1½ horsepower in archaic units). A 400-kg fully-loaded pod traveling at 28 m/s would have 154 kJ of kinetic energy, which would take 2.3 minutes to build up at 1120 watts, so trips of 5 minutes or more would reach full speed. A 4.6-minute one-way trip, then, would be 3.9 km; a half-hour trip (supposing a maximum of 28 m/s) would be 47 km, so a city organized around such a transport grid could grow to a diameter of 47 km before any trip was over half an hour, not counting the half-minute on each end to walk the 35 meters or less to the nearest station.
The tunnels, being underground, don’t create any noise or pedestrian hazards on the surface. If they’re built by trench-and-cover methods, they would be super cheap; the trench could be only 2 meters deep, or 5 meters to add crossover tunnels and in cases where it’s considered important to keep the surface free of noise and vibration. It’s necessary to have smaller drainage tunnels below so that seeping rainwater doesn’t fill up the tunnels. In areas where the water table may reach the level of the tunnels, despite the best attempts of drainage tunnels and sump pumps to lower it, bare dirt isn’t a viable option, and the tunnel floors would need to be at least dirt grouted with lime, or something like that. Also, unless the tunnels were actually waterproofed, they would be unusable when the tunnel was actually below the water table.
With regenerative braking, only the energy required to overcome friction would be lost; the energy to accelerate the pod is recovered when it slows down, using the same motor/generator that accelerated it.
Low-pressure 25 PSI (170 kPa) bicycle tires, like those used on beach-cruiser bicycles, are gentle enough on the ground to not crumble most soils. (Soil bearing capacities generally used in civil engineering range from 75 to 600 kPa, with “compact sand” being 300 kPa.) Supporting 400 kg on such tires would require a ground-contact patch of 0.023 m² or 230 cm² (35 in² in archaic units). If the tires are 5 cm wide, then you need a bit more than 46 cm of tire contact length, which is easily achievable with two or more tires.
When you enter a transport station, you specify your destination station, as with Uber Cab. If a pod is already available at that station, you just get in; otherwise, a pod is dispatched to you, and after you get in, it’s dispatched to your destination. Dispatching works by planning out a route using available tunnel segments, then reserving that route so that no other pod will use it until you’re done, to prevent collisions even if something goes wrong. Since there are plenty of tunnels (20 per kilometer, even without any stacking!), and the trips are so quick, it should be easy to build enough capacity that it’s practical to reserve the whole circuit end-to-end before starting.
The pods are long and low; users recline in a nearly-horizontal position, one behind the other, or with children in the laps of their caretakers. This allows the tunnels to be only 1½ meters tall. The pods aren’t collective vehicles like buses and trains — you can be alone, or share it with your family or friends going to the same destination. So you don’t need to make any stops in the middle to pick up or drop off other people, the way a bus or elevator does.
By contrast, Buenos Aires, where I live now, has the best public transport of any city I’ve ever lived in. To take a bus downtown, I need to walk three blocks (6 minutes) to the bus stop, wait an average of 5 minutes (but sometimes 10) for the bus, wait on the bus anywhere from 30 to 60 minutes to go the 4 km to downtown. So it takes, let’s say, 51 minutes on average, with an enormous variance. In the same amount of time, the tunnel PRT system could take you anywhere within 82 kilometers — not just downtown, but uptown too.
Consider a five-kilometer segment of a six-lane highway of cars as a point of comparison. It can accommodate about 1.5 cars per second in each direction, with about 1.2 people per car, for a total of 1.8 people per second per direction. When there’s no traffic jam, they can travel at 28 m/s, so their latency is about three minutes, although that doesn’t count the time to get on and off the highway and park. It’s about 20 meters wide, noisy, impassable to pedestrians and wildlife, requires constant attention from drivers, and needs a couple of meters of concrete and stones to support it.
What does the tunnel PRT system need to reach 1.8 people per second per direction over a three-minute route with end-to-end reservations?
If we pessimistically assume only one person per pod, it needs 100 tunnels in each direction. If the tunnels are stacked three deep (5 meters), that’s a width of about 1.7 km. That pod will weigh no more than 150 kg, rather than 400 kg, so it can reach full speed in 53 seconds, and needs another 53 seconds to decelerate. In that case, it will be able to travel 3.6 km in those three minutes, which is a bit shorter than the 5-km car highway, though not much. (Traveling the full 5 km requires another 50 seconds and consequently another 28 tunnels each way to reach the same throughput.)
If we instead have fully loaded 400 kg pods carrying four people each, it only needs 25 tunnels in each direction, or 830 meters of width, but in 1½ minutes, the heavier pods only reach 22 m/s, so they only cover 2.0 km in three minutes. Traveling the full 5 km requires 5½ minutes, and thus 150 tunnels in each direction. XXX calculation error
However, generally six-lane highways are few and far between in the urban landscape; parallel highways are usually on the order of 10 km apart. So this tunnel PRT system could actually support highway-like volumes and rapidity of travel everywhere, but without the noise, pollution, traffic jams, barriers, and parking problems, and possibly without even the danger, though safety issues are notoriously hard to predict.
As another comparison, consider the B line of the Buenos Aires Subte (our subway), which is 11.8 km long, which it traverses in 30 minutes, and the most popular line. It carried 6'035'183 passengers in January 2019, the latest month for which statistics are available, 195000 per day on average. About one-twenty-fifth of the monthly total across all lines is an average workday — the weekends are much less trafficked. As of 2017, 25% of the total across all lines is between 16:00 and 19:00, when the trains are jam-packed full. The city has adopted a goal of a train every 3 minutes during rush hours, although of course any irregularity in service can stretch the delay to many times that; with such short headways, even in the absence of other problems, bus clumping is likely.
If we apply the older systemwide averages to the current numbers for the B line, we estimate 240'000 passengers per workday, of which 60'000 are between 16:00 and 19:00, an average of 20'000 per hour, 5.6 per second. The use of the B line is concentrated toward downtown, and especially in the first seven stations, but we can pessimistically assume that usage is uniformly distributed over the whole line, giving an average trip length of one-third the length of the total, 3.9 km and 10 minutes. The tunnel PRT system outlined here would need 4.6 minutes for a 3.9-km trip, thus running twice as fast. Let’s pessimistically assume that all the passengers are going in the same direction (which is close to the truth at rush hour), requiring an equal number of empty return pods.
If you have four passengers per pod, you would need 760 PRT tunnels to provide this level of service. If these were spread out over a kilometer at 50-meter intervals, this would require tunnels stacked 38 levels deep, 57 meters! This is about twice as deep as the deepest part of the existing B line; the original trench-and-cover tunnel reached its deepest point at 17 m, but the newer stations are deeper.
(Alternatively, you could just put the tunnels next to each other with only the occasional space in between for a station, which would spread them out over a kilometer and a half; or go for in-between measures, like three-tunnel-deep stacks a couple of meters apart.)
At the other extreme, if you have one passenger per pod so that it weighs only 150 kg, the trip would be quicker (3.2 minutes), but you’d need 3000 tunnels.
Because the trips are so much shorter and the pods are not linked up into inflexible trains, the total number of pod seats needed to deliver this level of service is about an order of magnitude lower than the number of subway train seats; pods totaling only a few thousand seats would be sufficient to provide hundreds of thousands of rides per day.
The scaling laws for invisible thoroughfares like those described above are perfect for constant-speed vehicles: to go twice as far with end-to-end reservations, you need twice as many tunnels, which means the people are traveling distributed over a thoroughfare area twice as wide, so the thoroughfare has a constant aspect ratio regardless of its length. However, by including occasional stops — say, every 10 minutes for single-person pods, or every 30 minutes for fully-loaded pods — the amount of traffic can scale arbitrarily high. This more or less corresponds to what we would call a “headway” of 10 or 30 minutes in a traditional mass-transit system.
Putting 28 amps on the 40-volt DC cables is probably best achieved with a higher-voltage distribution system (insulated, perhaps buried, and perhaps AC) stepped down and rectified if necessary with a buried transformer every few hundred meters; if you try to do the distribution directly over kilometers at 40 volts, the cable losses will be high.
For comfort as well as limiting floor and tire wear, you’d probably want to limit accelerations to, say, half a gee, but with such small motors, this is only a problem at the very beginning and end of the trip; once you’re up to 1.5 m/s (310 ms at half a gee), the 1120-watt power limit is more tightly constraining than the half-a-gee limit, even at 150 kg. (At 400 kg, the transition happens even sooner, at 570 mm/s, 116 milliseconds into the trip.)
You’d want to tilt for curved tunnels; this is easily accommodated, even for widely variable speeds, if the tunnels are circular in cross-section and the pod wheels are placed so as to touch this circle. The tightness of the curves at a given speed is limited by the centripetal acceleration — both for passenger comfort and to limit tire loading, you want to limit that. Centripetal acceleration is just v²/r, so if you limit it to half a gee as well, the curved tunnels for changing direction at maximum speed should have a radius of curvature of 160 meters or more ((28 m/s)²/(4.9 m/s²)); to limit jerk, the tunnel curvature should change smoothly into and out of the curve.
Alternatively, you could use tighter-radius curves to change direction at lower speeds; for example, at the beginning and end of the trip.
Since such a system would enormously reduce the cost of living in lower-density “suburbs”, we can expect that people would opt to move to areas of lower populational density, though perhaps not proportionally. Because it would eliminate the usefulness of cars within its ambit, walking would become much safer and more pleasant. We can guess that this would motivate people to optimize their neighborhoods for walkability, making them more parklike, and perhaps to increase the distances between potentially dangerous industrial facilities and personal spaces (such as houses) or public spaces (such as shops).