Question

Five-minute average speed in km/h and flow rate in veh/h during 12 hours were collected from a section of the eastbound lanes of Highway 401 in Toronto. The collected data were plotted to form the scatter diagram shown in the figure below. A curve was drawn to describe the general relationship between speed and flow rate q as shown in the figure. 120 300 Speed km/h 60 1000 Flow Rate vel (a) From the above speed-flow rate curve, determine the free-flow speed, critical speed S, and capacity of the freeway (2 marks) (b) Calculate the critical density D. (2 marks) (c) Calculate the jam density D, if you know that an increase of 1 vehicle per km will reduce speed by 2 km/h. (2 marks) (d) Sketch the relation between speed and density. Make sure to show free-flow speed, critical speed, critical density and jam density on the graph. (2 marks) (e) Sketch the relation between flow rate and density. Make sure to show free-flow speed, critical speed, critical density, jam density and capacity. (2 marks)

Answer 1

a) from the above graph speed increases from zero upto its critical flow speed speed (which happens when flow is maximum) and then speed drops

So Free Flow Speed = 100 kmph

critical speed S₀ = 80 kmph

Capacity = 2000 veh/hour

b) Critical density = 2000/free flow speed = 2000/100 = 20 veh/km/lane

c) According to greenshields model

* K,

where where u_f is the free-flow speed, and k_j is the jam density.

so at jam density are given flow rate = 1 *2 = 2 veh/hour

so 2000 = k_jam (80 - 80²/100)

so we get k_jam = 2000/16 = 125 veh/km

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General graph of speed versus density is a straight line like below

k=20 Speed vs Density uvsk = 125 n. paads Kjom Density -

General graph of flow versus density is shown below

Flow vs Density qvs k > capacity e critical spoed Flow (9) >Kjam= 125 Density (k) free flow steed

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