Highway Engineering

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4.2.2 Flow-density relationship


Combining Equations 4.5 and 4.6, the following direct relationship between flow
and density is derived:

(4.9)


This is a parabolic relationship and is illustrated below in Fig. 4.2.
In order to establish the density at which maximum flow occurs, Equation 4.9
is differentiated and set equal to zero as follows:

since ufπ0, the term within the brackets must equal zero, therefore:

(4.10)


km, the density at maximum flow, is thus equal to half the jam density,kj. Its
location is shown in Fig. 4.2.

1


2


0


2


-=


=


k
k

k

k

m
j

m

j

, thus

dq
dt

u k
f kj

=-


Ê


ËÁ


ˆ


̄ ̃


1 2 = 0


quku

k
k

k

quk

k
k

f
j

f
j

== -


Ê


ËÁ


ˆ


̄ ̃¥


=-


Ê


ËÁ


ˆ


̄ ̃


1


2

, therefore

76 Highway Engineering


(^00)
qm
km kj
Density (veh/km)
Flow (veh/h)
Figure 4.2
Illustration of flow-
density relationship.


4.2.3 Speed-flow relationship


In order to derive this relationship, Equation 4.6 is rearranged as:

(4.11)


By combining this formula with Equation 4.5, the following relationship is
derived:

kk u
j uf

=-Ê


ËÁ


ˆ


̄ ̃


1

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