33.2.1 - Relationship between Speed and Density
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Introduction to Speed-Density Relationship
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Today, we will discuss the relationship between traffic speed and density, as introduced by Greenshield’s model. Can anyone tell me what speed and density represent in traffic terms?
Speed is how fast the vehicles are moving.
Density is the number of vehicles per unit length of road.
Exactly! Speed is the rate at which vehicles travel, while density refers to how crowded the road is. Now, Greenshield proposed a linear relationship between these two. What do you think that means?
It means as one increases, the other decreases, right?
Yes, that’s correct! As the density increases, the speed tends to decrease. This relationship is represented by the equation v = vf (1 - k/kj). Remembering this can be easier with the mnemonic 'Very Free, Keep Jammed'.
What does vf and kj represent again?
Great question! vf is the free flow speed when density is zero, and kj is the jam density where vehicles cannot move. Let's summarize: The denser the traffic, the slower the speed. Keeping this in mind helps manage traffic effectively.
Calculating Flow
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Now let's delve into how we can derive traffic flow from speed and density. Using the equation, who can tell me the flow equation?
Is it q equals k times v?
Exactly! q = k * v. This means flow is the product of vehicle density and mean speed. If we substitute Greenshield's speed equation into this, what happens?
It gives us a new expression for flow related to density?
Correct! We get a parabolic relationship for flow as a function of density. Remembering the shape can help understand traffic behavior better – think of it like a hill where there's an optimal peak of flow.
So flow increases to a peak and then decreases?
Exactly! This peak represents maximum flow. And to find this maximum flow, we need to know our free flow speed and jam density.
Understanding Maximum Flow
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Let's now focus on maximum flow. So, based on our understanding, how do we determine the density at maximum flow?
Doesn't it involve differentiating the flow equation and setting it to zero?
Exactly! By doing that, we find that the density for maximum flow is half the jam density. What does that mean for practical traffic management?
It helps to know how much traffic can flow before it starts to congest, right?
Exactly! Knowing when to expect maximum flow allows us to manage traffic systems more efficiently. And remember, the formula for maximum flow is q_max = (vf * kj)/4.
So if we have our vf and kj, we can easily find the maximum flow?
Yes! Always use those parameters when planning or analyzing traffic flows. Let's recap: density at maximum flow is halfway to jam density, and maximum flow depends on both free flow speed and jam density.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Greenshield's model describes a linear relationship between traffic speed and density, presenting equations that illustrate how flow can be derived from these parameters. The section discusses definitions for free flow speed, jam density, and maximum flow, along with methods to calculate these values.
Detailed
Relationship between Speed and Density
This section focuses on the fundamental relationship between traffic speed and density, primarily using Greenshield's macroscopic stream model. The model establishes a linear relationship wherein:
- Key Parameters:
- Mean speed (v): Speed of vehicles at varying densities (k).
- Free flow speed (vf): Maximum speed when traffic density is zero.
- Jam density (kj): Maximum density where vehicles cannot move.
- Greenshield’s Model:
-
The model is defined by the equation:
$$v = vf (1 - \frac{k}{kj})$$
This indicates that as density increases (approaching jam density), speed decreases. - Flow Relationships: The relationship between traffic flow (q), speed, and density is derived as follows:
- Flow is expressed as:
$$q = k imes v$$ - By substituting Greenshield's equation, a parabolic relationship between density and flow emerges.
- Maximum Flow: The text explains how maximum flow, related to maximum density, can be calculated:
- The maximum density for maximum flow is given by:
$$k_0 = \frac{kj}{2}$$ - The maximum flow itself can be derived as:
$$q_{max} = \frac{vf\times kj}{4}$$
This highlights that maximum flow is achieved at half of the free flow speed.
Understanding this relationship is crucial for traffic management, as it helps predict traffic conditions and optimize flow on roadways.
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Greenshield's Linear Relationship
Chapter 1 of 4
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Chapter Content
Greenshield assumed a linear speed-density relationship as illustrated in figure 33:1 to derive the model. The equation for this relationship is shown below.
v
f
v =v .k (33.1)
f
− k
[j]
where v is the mean speed at density k, v is the free speed and k is the jam density. This equation ( 33.1) is often referred to as the Greenshields’ model. It indicates that when density becomes zero, speed approaches free flow speed (ie. v f when k 0).
Detailed Explanation
Greenshield's model proposes a simple linear equation relating speed (v) and density (k) of traffic. The important parameters are:
1. Mean Speed (v): The average speed of vehicles at a certain density k.
2. Free Speed (v_f): The maximum speed when there are no vehicles on the road (density = 0).
3. Jam Density (k_j): The highest density at which vehicles can still move (maximum congestion).
At a density of zero, vehicles can travel at free speed, but as density increases, speed decreases linearly until they reach jam density where movement becomes very slow.
Examples & Analogies
Imagine a highway with no cars. Drivers can go as fast as they want - this is free speed. As more cars join the highway, everyone has to slow down because there isn’t enough space for everyone to travel at high speeds. In the extreme case where every lane is completely full, cars are at jam density and can barely move. This is similar to what happens in a traffic jam, where the relationship between the number of cars (density) and how fast they can go (speed) is linear until they can't go any faster.
Flow-Density Relationship
Chapter 2 of 4
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Chapter Content
Once the relation between speed and flow is established, the relation with flow can be derived. This relation between flow and density is parabolic in shape and is shown in figure 33:3. Also, we know that
q =k.v (33.2)
Detailed Explanation
After understanding how speed and density relate, we can derive the relationship involving flow (q). The flow is calculated by the product of density (k) and speed (v). This relationship demonstrates that as density increases, the flow initially increases, reaches a maximum, and then decreases, forming a parabolic curve.
Examples & Analogies
Imagine pouring water into a funnel. Initially, as you pour in slowly (low density of water), the water flows out smoothly (increasing flow). If you pour too much too quickly (high density), the funnel can't handle it anymore, and the flow decreases (decreasing flow). This shows that there is an optimal 'density' for maximum flow.
Deriving Maximum Flow
Chapter 3 of 4
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Chapter Content
To find the density at maximum flow, differentiate equation 33.3 with respect to k and equate it to zero. ie.,
dq/dk = 0
v
f
v .2k = 0
f
− k
j
k
j
k = 2
Denoting the density corresponding to maximum flow as k_0,
k
j
k = (33.5)
0
2
Detailed Explanation
To find the density where traffic flow is maximized, we apply a method used in calculus called differentiation. We take our earlier equation (related to flow) and differentiate it to find where it peaks. This maximum flow density, denoted as k_0, occurs at half of the jam density (k_j), showing that there is a specific point before reaching maximum congestion where traffic can flow best.
Examples & Analogies
Consider a highway merging from multiple lanes into two lanes. If the merging happens too slowly, vehicles start to back up and flow decreases. The best density of cars allows for a smooth transition without bottlenecks, just like a carefully timed relay race, where the smooth changes between runners allows maximum performance.
Maximum Flow Equation
Chapter 4 of 4
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Chapter Content
Substituting equation 33.5 in equation 33.3,
k v k^2
j f j
q_max = v . .
f 2 − k^2
j [ ]
Thus the maximum flow is one fourth the product of free flow and jam density.
Detailed Explanation
To calculate the maximum flow, we substitute k_0 (the density at maximum flow) into our previously derived flow equation. It turns out that the maximum flow (q_max) is a quarter of the product of free speed (v_f) and jam density (k_j). This gives us a precise mathematical relationship that helps predict how many vehicles can pass a point in a given time under optimal conditions.
Examples & Analogies
Think of a lemonade stand at a fair. If more customers come than you can serve (jam density), even if they want lemonade (flow), your ability to serve them (speed) suffers. At the optimum point, you can serve the most customers effectively without running out of inventory, ensuring maximum satisfaction.
Key Concepts
-
Greenshield's Model: Establishes a linear relationship between speed and density.
-
Free Flow Speed (vf): Max speed when density is zero.
-
Jam Density (kj): Maximum density where traffic stops.
-
Flow (q): The product of density and speed.
Examples & Applications
In a traffic simulation, if the free flow speed is 60 km/h and the jam density is 200 vehicles/km, the maximum flow can be calculated as 60 km/h * 200 / 4 = 3000 vehicles/hour.
If the density of traffic on a road is observed to be 50 vehicles/km, substituting this into Greenshield's model can help estimate the expected speed.
Memory Aids
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Rhymes
Speed high and density low, on the road let traffic flow.
Stories
Imagine a busy highway. When traffic is light, cars zoom past like a breeze. But as more vehicles arrive, speeds slow down, until suddenly, nobody can move—this is jam density!
Memory Tools
Remember 'Space Makes Free'—more space (lower density) allows for free flow (higher speed).
Acronyms
SPEED
'S' for Speed
'P' for Peak flow
'E' for Equations
'E' for Efficiency
'D' for Density.
Flash Cards
Glossary
- Mean Speed (v)
The average speed of vehicles at a given traffic density.
- Free Flow Speed (vf)
The speed of vehicles when the density is zero, representing optimal flow conditions.
- Jam Density (kj)
The maximum density of vehicles where traffic comes to a complete halt.
- Traffic Flow (q)
The number of vehicles passing a point in a given time, expressed as a function of density.
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