Highway Engineering

(4.12)

This relationship is again parabolic in nature. It is illustrated in Fig. 4.3.

qku

u

j uf

=-ÊËÁ ˆ ̄ ̃

2

Basic Elements of Highway Traffic Analysis 77

Flow (veh/h) qm

um

Speed (km/h)

Figure 4.3

Illustration of speed-

flow relationship.

In order to find the speed at maximum flow, Equation 4.12 is differentiated
and put equal to zero:

since kjπ0, the term within the brackets must equal zero, therefore:

(4.13)

um, the speed at maximum flow, is thus equal to half the free-flow speed,uf. Its
location is shown in Fig. 4.3.
Combining Equations 4.10 and 4.13, the following expression for maximum
flow is derived:

therefore

(4.14)

q

uk

m

= fj

4

quk

uk

mmm

=¥= ¥fj

22

1

2

0

2

-=

=

u

u

u

u

m

f

m

f

, thus

dq

dt

k

u

j uf

=-Ê

ËÁ

ˆ

̄ ̃

1 =

2

0