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DEPARTMENT OF CIVIL ENGINEERING

Transport and Traffic Engineering

(With Answers)

Practical Class 2: Traffic Flow Fundamentals and Traffic Flow Theory

The exercises in this practical class examine different types of fundamental diagrams in traffic flow. These exercises draw on the material presented in the lecture slides “Traffic Flow Fundamentals 2” and “Fundamental diagrams’’.

You will probably need a spreadsheet such as Excel to complete these exercises,
remember to bring a computer with spreadsheet software into the practical class.

EXERCISE 5

Part 2

Two aerial photographs were taken, 30 seconds apart, over one east-bound lane of a freeway. The following results were recorded.

Vehicle Position in Photo 1 (m) Position in Photo 2 (m) Δx (m) Speed (km/h)
1 609 897 288 34.56
2 640 814 174 20.88
3 579 792 213 25.56
4 457 732 275 33
5 152 497 345 41.4
6 0 305 305 36.6

Note: The vehicle positions (in metres) are taken from a single reference location.

Question 17a: Plot these trajectories on a sheet of paper (or in a spreadsheet) and compute the average flow, density and space-mean speed over the 900 metre length of the lane.

Space mean speed is the numerical average of the speeds of the six vehicles shown on (either of) the aerial photographs:

= AVERAGE(34.56, 20.88, 25.56, 33, 41.4, 36.6).

Note that at every fixed time (0-30s), the speed of vehicle i (i=1,…,6) is constant. The space mean speed is calculated by the observations obtained at a fixed time, so the mean formula is used.

The Space-mean speed is 32 kilometres per hour

We need to assume that these six vehicles are the only ones visible.
Over the length, L= 900 m, there are N=6 vehicles

Density, k = N / L = 6/900 = 0.00667 veh/m

The Density is 6.67 vehicles per kilometre

[Grab your reader’s attention with a great quote from the document or use this space to emphasize a key point. To place this text box anywhere on the page, just drag it.] Note: the density is observed at a fixed time over a specified length (900m). At any point of time, we see 6 vehicles within

900m road section.

From the Fundamental relationship, q = kvs

Flow = Density × Space Mean Speed

Flow = 6.67 × 32 = 213 veh/h

The Average Flow is 213 vehicles per hour

Note: the flow is observed at a fixed point over a time period (30s)

Question 17b

Time (s) 1st car 2nd car 3rd car 4th car 5th car 6th car 7th car vs (m/s) =mean (v1:v7)
1 11.9024 13.1155 12.5517 13.3015 13.716 13.6459 13.716 13.13557
3 11.6129 12.7437 13.079 13.1186 13.7099 13.7008 13.716 13.09727
5 10.8936 11.8476 13.271 12.5639 13.716 13.716 13.716 12.81773
7 10.5522 11.11 13.274 12.2133 13.6947 13.716 13.716 12.61089
9 10.6436 10.7594 13.018 12.189 13.3411 13.716 13.716 12.4833
11 10.671 10.6589 12.4176 12.192 12.9357 13.716 13.716 12.3296
13 10.6345 10.665 12.1128 12.1676 12.9479 13.716 13.6703 12.27344
15 10.2474 10.665 12.1707 12.0731 12.9875 13.6764 13.6459 12.20943
17 9.5494 10.6497 12.192 12.1097 12.9784 13.6246 13.7008 12.11494

EXERCISE 6

Data collected on a metropolitan freeway was used to find that speed (v in km/h) and density (k in veh/km) were related by the following expression:

v = 100 – 0.7k

Question 18: What is the free-flow speed on the freeway?

Free-flow speed (vf) is the greatest possible speed, occurring when there are no other vehicles on the road (i.e. density, k = 0)

Substitute k = 0 into the given equation

vf = 100 – 0.7× 0 = 100

The Free-flow Speed is 100 km/h

Question 19: What is the jam density?

The jam density (kjam) is the greatest possible density, occurring when the road is completely congested and traffic is stationary (i.e. speed, v=0)

Substitute v = 0 into the given equation

0 = 100 – 0.7× kjam

kjam = 100 ÷ 7 = 142.8

The Jam Density is 143 vehicles per kilometre

Question 20: What is the jam spacing (average spacing between fronts of stationary vehicles)?

This Jam Density of 143 metres corresponds to a jam spacing of 1000 ÷ 143 = 7.0 m/veh)

The Jam Spacing is 7 metres per vehicle

Question 21: What is the formula for the fundamental relationship, expressing flow
(q in veh/h) in terms of density (k in veh/h)?

Flow = Density × Speed

Substitute the above relationship for speed as a function of density into this equation.

v = 100 – 0.7k

q = 100k – 0.7k2

This is the equation for a parabola

Question 22: What is the maximum flow that can be accommodated on the freeway?

The maximum flow, qcap occurs at a value of density (kcr) when flow does not change with changing density, i.e. dq/dk = 0

dq/dk = 100 – 0.7×2k

100 – 1.4kcr = 0

kcr = 71.43 veh/km

Substitute into the equation for the parabola

qcap = 100kcr – 0.7 kcr 2 = 100 × 71.43 – 0.55 × (71.43)2 = 3571 veh/h

Alternatively, qcap = vf× kjam ÷ 4 = 100 × 142.8 ÷ 4 = 3571

The Maximum Flow, qcap, is 3571 vehicles per hour

Question 23: At what speed would vehicles be travelling to give that maximum flow?

The critical speed, vcr is the speed at which flow is a maximum (3571 veh/h) and density is the critical density (71.43 veh/km).

Speed = flow / density = 3571 / 71.43 = 50

The Critical Speed, vcr, is 50 kilometres per hour

Alternatively, we can substitute k=kcr into the original equation relating speed and density

vcr = 100 – 0.7× 71.43 = 50

Question 24: Plot the relationship for pace (t1km in seconds to travel 1 kilometre) in terms of density (k in veh/km)

Speed / Density (v/k) relationship

Flow/ Density (q/k) relationship – the Fundamental Diagram

Rate of change of Flow vs Density – This is used to find maximum flow and critical density

Speed / Flow (v/q) relationship

The time taken to travel 1 kilometre is the inverse of the speed

EXERCISE 7

Data collected on a metropolitan freeway was used to find that speed (v in km/h) and density (k in veh/km) were related by the following expression:

Question 25: What is the free-flow speed on the freeway?

Free-flow speed (vf) is the greatest possible speed, occurring when there are no other vehicles on the road (i.e. density, k = 0)

Substitute k = 0 into the given equation

vf = 100 × (2 + 0.008 × 0 – e0.008×0) = 100 × (2 + 0 – 1) = 100

The Free-flow Speed is 100 km/h

Question 26: What is the jam density?

(Hint: You may need to use a numerical solver such as Solver in Excel)

The jam density (kjam) is the greatest possible density, occurring when the road is completely congested and traffic is stationary (i.e. speed, v=0)

Substitute v = 0 into the given equation

We need to solve this numerically, for example using Solver in Microsoft Excel, to give:

kjam = 143.3 veh/km

The Jam Density is 143.3 vehicles per kilometre

Question 27: What is the jam spacing (average spacing between fronts of stationary vehicles)?

This Jam Density of 143.3 m corresponds to a jam spacing of 1000 ÷ 143.3 = 6.97 m/veh)

The Jam Spacing is 6.97 metres per vehicle

Question 28: What is the formula for the fundamental relationship, expressing flow
(q in veh/h) in terms of density (k in veh/h)?

Flow = Density × Speed

Substitute the above relationship for speed as a function of density into this equation.

Question 29: What is the maximum flow that can be accommodated on the freeway?

(Hint: You will need to use a numerical solver such as ‘Solver’ in Excel)

The maximum flow, qcap occurs at a value of density (kcr) when flow does not change with changing density, i.e. dq/dk = 0

We need to solve this numerically, for example using ‘Solver’ in Microsoft Excel, to give:

kcr = 86.64 veh/km

Substitute this into the equation for flow as a function of density:

qcap = 6005 veh/h

Alternatively, we could one again use a numerical method such Solver in Microsoft Excel to find the maximum value of flow, q, by varying the density, k.

The Maximum Flow, qcap, is 6005 vehicles per hour

Note that the formula qcap = vfkjam/4 no longer is valid because we do not have a parabola.

Question 30: At what speed would vehicles be travelling to give that maximum flow?

The critical speed, vcr is the speed at which flow is a maximum (6005 veh/h) and density is the critical density (86.64 veh/km).

Speed = flow / density = 6005 / 86.64 = 69.3 km/h

The Critical Speed, vcr, is 69.3 kilometres per hour

Alternatively, we can substitute k=kcr into the original equation relating speed and density

vcr = 69.3

Question 31: Plot the relationship for pace (t1km in seconds to travel 1 kilometre) in terms of density (k in veh/km)

Speed / Density (v/k) Relationship – this is no longer a straight line

Flow / Density (q/k) Relationship – this is no longer a parabola

Rate of change of Flow vs Density – This is used to find the maximum flow and the critical density. It is no longer straight line and must be solved numerically

Speed / Flow (v/q) Relationship – this is no longer a sideways parabola

The time taken to travel 1 kilometre is the inverse of the speed

EXERCISE 8

A section of highway is known to have a free-flow speed of 75 km/h and a capacity of 3000 veh/h. In a given hour, 2000 vehicles were counted at a specified point along this highway section.

Question 32: Given that the linear speed-density relationship applies, what would you estimate the space-mean speed of these vehicles to be?

The linear relationship between speed and density is:

We are given vf = 75 km/h and qcap = 3000 veh/h

For this linear speed-density relationship, we can use

So kjam = 4 × 3000 ÷ 75 = 160 veh/km

Inverting the linear speed density equation gives

Multiplying by the fundamental equation q = kv gives

Substituting in the given value of vf= 75 and the calculated value of kjam = 160 gives

This is a quadratic equation that can be solved for q = 2000 veh/h

Solve for v = 15.85 or v = 59.15

Check: when v = 15.85, k = 126.2, q = kv = 2000

Or, when v = 59.15, k = 33.8, q = kv = 2000

The space-mean speed of these vehicles is either 15.85 km/h, or 59.15 km/h

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