Traffic Flows (2)

Referring back to 15/03/2009 entry 'Traffic Flows', I've been considering the consequences of merging lanes of traffic, by analogy with the Bernoulli Equation.

The Bernoulli Equation in fluid dynamics, as it relates to the flow of an incompressible fluid, can be expressed as:

Where v is the fluid flow speed (in a straight line), g is the acceleration due to gravity, z is the elevation of the point under consideration above a reference plane, with the z direction opposing g, p1 is the pressure of the fluid at the point under consideration and p2 is the density of the fluid.

Taking this equation to be analogous to traffic flows, both g and z cease to relate to any meaningful physical concept, and the equation can be simplified by discounting them.

Continuing to refine the analogy, the density, p2 of the fluid may be considered to correspond to the number of vehicles per metre of lane (though perhaps we should be employing yards as our unit of measure here, in deference to the requirements of the Road Traffic Act). The Bernoulli equation holds true for an incompressible fluid, and avoidance of the 'compression' of traffic (thereby increasing its density) is precisely what we seek to achieve.

p1, the pressure of the fluid, is a slightly more nebulous concept in traffic flow terms, corresponding essentially to the extent to which motorists are compelled by obstruction to travel more slowly than the nominal speed-limit for the road allows.

Defining pressure in terms of vehicles travelling along a road is challenging, but I propose to use the 'mean free path' (MFP) concept from Collision Theory as the basis for a model.

The MFP for a gas can be defined as follows:

where kB is the Boltzmann constant, T is temperature, p is pressure, and d is the diameter of the gas particles.

On the (hopeful) assumption that vehicles on a road do not actually collide, the MFP may nonetheless be considered to correspond to a measure of the freedom of manoeuvre enjoyed by the individual motorist (which is of course dependent on the weight of traffic).

d for these purposes may be defined as the width of a lane (as multiple vehicles cannot occupy a single lane at the same point along the road. Since the lane width is a constant we shall ignore it. kB, pi and root2 are constants of proportionality, which we may ignore. and the temperature features in this equation because it affects the velocity of the individual particles. Since temperature does not affect driving speed we shall ignore it too.

Effectively then, the pressure experienced by the traffic is inversely proportional to the MFP, which itself is proportional to the weight of traffic, or the length of lane per car.

p1/p2 thus becomes (length of lane per car)/(cars per length of lane), which thus becomes (length of lane per car)^2

Overall then, 1/2(v^2) + (length of lane per car)^2 = constant.

We have established that in order to prevent congestion we must prevent a reduction in the length of lane per car.

Rearranging the above equation thus gives:

length of lane per car = (constant - 1/2(v^2))^(1/2).

Where two lanes merge to one, the number of cars per lane doubles (obviously). To compensate for this effect, (and thus prevent an increase in traffic density), the velocity of the traffic must increase, so as to effectively double the length of lane per car (as it would have been had the lanes not merged)

Let velocity of the traffic in the two lanes = v(1), velocity of the traffic merged to one lane = v(2) and length of lane per car = L,

L = (constant - 1/2(v(1)^2))^(1/2)

2L = (constant - 1/2(v(2)^2))^(1/2)

This rearranges such that v(2) can be defined as (4v(1)^2 - 3(constant))^(1/2). The value of the constant can presumably be determined by observation, but it is necessarily not negative. Note also that 3(constant) cannot exceed 4v(1)^2 without resort to imaginary numbers.

This equation tends ever closer towards v(2) = 2v(1) as v(1) increases. To a first approximation then, v(2) = 2v(1) may be used as a guide in determining the speed limit change required to prevent traffic congestion at a point where lanes merge, i.e., a dual carriageway with a speed limit of 60 mph should merge to a single lane with a speed limit immediately beyond the merge-point, of 120 mph.

The MFP is considered on the hopeful assumption that vehicles don't actually collide...