Wednesday, May 9, 2012

How Much, Really, Can You Squeeze In?

This post is inspired by a quote from a Mayor's Office of Transportation and Utilities (MOTU) spokesperson in an interview from Plan Philly having to do with the 10th Street Chinatown bike lane: that the technical maximum per traffic lane is 800 vehicles/hour.

Now, I'll agree that there's a technical maximum. I am not so sure that it is so blasély quantifiable. Think about it: the movement of a car has two geometrical elements that must pass any point or line before the next can take its place in safe progression--the carbody itself, as well as the reaction space in front of the car. This unoccupied reaction space grows proportionally to the speed of travel--it decreases the total number of vehicles which can safely progress through a point the faster these vehicles are going.

Another way of saying this is that per-lane traffic progression (throughput) is a function of the reaction space of the car, itself a function of the speed of the car. As mean speed goes up, throughput (how many cars can actually progress through a given point) goes down, and vice versa, until stop is achieved and the system is clogged.

Ironically enough, peak throughput occurs when the system is so congested as to inhibit unimpeded movement, but no so congested as to actually be clogged--that is to say, a rush hour traffic jam. Peak throughput occurs in speeds that utterly fail to capitalize on the car's major advantages (i.e. its speed premium) and so autocentric traffic systems are designed in such a way as to avoid peak throughput. This, in its turn, incurs further costs (transportation mode restriction, increased spatial consumption = sprawl, etc.).

Let us assume mean speed on the stretch of 10th St. in question is 10 mph. It's in Chinatown, which lots of stoplights, jaywalkers, and cars coming from the north or off the Vine Street Expwy., so this is a reasonable assumption. Given the throughput model I've just outlined here, how many cars can pass per hour in a single driving lane?

Since 1 mile = 5280 ft., 10 miles = 52,800 ft. Therefore a vehicle traveling 10 mph traverses 52,800 feet in an hour. The PA Driver's Manual recommends keeping 6 seconds' stopping distance, and a rule of thumb is that the stopping distance is roughly 1 carlength per 10 mph.

6 seconds' stopping distance is 52,800/60 = 800 feet/min, 800/60 = 13.3 ft/sec, 13.3*6 = 80 ft. stopping distance. Obviously not a tenable figure for any reasonable calculation*.

So let us use the rule of thumb then. Picture a row of cars lined up, bumper-to-bumper, exactly a mile long. Since the rule of thumb is that a car's stopping distance is itself every 10 mph, we can divide to obtain the number we're looking for.

For this purpose, let us say the average car is 18 ft. SUVs and light trucks are 20 ft. (or longer), while coupes are around 15 ft. This means there are 294 cars to the mile. Divide by half and we see that 147 cars can traverse 1 mile at 10 mph; multiply this by 10 and the number is 1,470 cars/hr (quite a bit more than 800 cars/hr)**.

Having tried these two calculations, one significantly lower than the MOTU number, and the other significantly higher--but the MOTU number floating serenely almost at the exact median between the two--we're forced to wonder: how did they come up with this number in the first place?
* But if you must know, this works out to an average geometric car being 98 feet long! It would take 7.37 seconds for this body to traverse that distance at that speed, which works out to 488.47 of those things in an hour--which is significantly lower than the MOTU number^.
^ Let's take this model further. 5280 ft/mile =  316,800 ft traversed at 60 mph. 316,800/3600 = 88 feet traversed per second. 88(6) = 528 ft reaction space + 18 ft (car) = 546 ft. geometrical object. 316,800/546 = 580.22 vehicles/hr at 60 mph^^^.
** According to this model, a car traveling at 20 mph is 3 vehicle units, at 30, 4, at 40, 5, and so on^^. So you can amortize to the hour to figure out traversal per point in an hour^^^.
^^ The model can be described as x = (C/y)z, where x is maximal cars/lane/mile in unimpeded conditions, C is carlengths per mile (a unit of length), y is geometric carlengths needed for free-flowing conditions per x10 mph, and z is mph.
^^^ But this contradicts the hypothesis I described earlier in the post. Throughput at 60 mph would be (297/7)60 = 2520 cars/lane/hr, which is noticeably higher than 10 mph throughput of 1470 cars/lane/hour. What do we make of this? A solution may be that the numbers we're using all describe free-flowing (unimpeded) conditions, whereas congestion is congested (impeded) conditions.

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