Question

QUESTION THREE On an urban highway, the relationship between vehicle speed and density can be represented by the Greenburg Model as the data in Table 3.1. Due to the roadwork on this highway, the width of both traffic lanes is narrowed and a bottleneck to traffic flow forms. The maximum flow on the unobstructed highway is 5000 vehicles per hour whilst on the section under repair the maximum flow is 4000 vehicles per hour. When the traffic flow approaching the roadwork is 4500 vehicle per hour, Calculate (a) the speed of the traffic stream at the maximum flow. (b) the speed of the traffic stream at approaching the roadwork before the commencement of the bottleneck. (c) the speed of the shockwave formed by the bottleneck. Table 3.1: Speed and Density observation at an urban highway 40 38 Speed, u Density,k km/hr vehicle/km 53 14 48 19 45 23 29 37 33 35 34 39 27 41 20 45 17 48 15 53 13 56 11 63 8 73 TABLE: 3.1

Answer 1

So we are given the data as follows

so we can plot the Speed-Density graph as follows

Problem says adopt greenshield model based on above graph

Greenshields model states that

v=A-B*k

Where:
v = speed (miles/hour, kilometers/hour)
A,B = constants determined from field observations
k = density (vehicles/mile, vehicles/kilometer)

So lets fit a straight line Regressionequation to speed density

I dont have data analysis pack in excel, so im finding the regression line by hand

Let our regression equation is given by Y = A + BX

where Y = dependent variable = speed(kmph)

X = independent variable = density (vpkm)

A and B are constants, A is the slope of the line and B is the intercept

So in such a case

B = r * Std. Dev(y)/Std Dev(x)

where r = pearssons coefficient and calculated as follows

so we get r = -0.97159

B = -0.85784

and A = mean(y) - B(mean(x)) = 63.957

So our greenshield equation for speed density i.s given by

Speed (U) = 63.957 - 0.858* Density(K)

Flow = Speed * density = (63.957 - 0.858* K)* K = 63.957K - 0.858K²

So we can plot our speed-density graph based on the FITTED VALUES

So we know that at jamming conditions

Speed_jam = 0 kmph, Density_jam = 74.542 vpkm

Freeflow speed = 63.957 kmph

So for maximum flow, different flow w.r. density

we get 63.957 - 2*0.858K = 0

so K for maximum flow = 63.957/1.716 = 37.27 vpkm

Speed at maximum flow = 63.957 - 0.858*K = 31.98 kmph

Maximum flow = 1191.89 vph

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b) before start of bottle neck flow = 4500 vph

this represents free flow conditions

So speed before commencement of bottleneck= free flow speed = 63.957kmph

density of vehicles

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c) Shockwave due to bottle neck

speed of shockwave at time of bottle neck = (Flow_{before repair} - Flow_bottleneck)/(density_{before repair} - density_bottleneck)

Flow before repair = 1200, flow_{at bottleneck} = 0

density before repair = 37.27, density_repair = jam density = 74.542 vpkm

so speed of shockwave = 1200/(37.27 - 74.542) = -32.19 kmph