Under what flow conditions is the above model valid?
Assuming that the expression:
can be used to describe the speed-density relationship of a highway, determine the
capacity of the highway from the data below using regression analysis.
What will be an ideal response?
First, to use linear regression, the equation given in the problem must be
converted to the form y = a + bx; this can be done by taking the natural logarithm
of each side of the equation as follows:
uf and kj are constants; therefore, in the regression,
a = ln(uf); b = –1/kj; x = k; and y = ln(us)
The only transformation needed for the input values is:
The linear regression analysis can be performed using equations 6.28, 6.29, and
6.30, or using a computer software spreadsheet package. Linear regression
analysis yields values of a = 4.0626 and b = –0.0103.
a) mean free flow speed, uf
uf = ea = e4.0626
uf = 58.1 mi/h
b) jam density, kj
kj = –1/b = –1/–0.0103
kj = 97.0 veh/mi
c) capacity, qmax
Capacity occurs at maximum flow. State flow in terms of density.
Take the derivative and set equal to zero to maximize flow; solve for density.
Take the natural logarithm of each side of the equation as follows:
ln(q) = (ln 58.1) – k/97.0 + ln(k)
Then take the derivative and set equal to zero.
1/q = –0.0103 + 1/k
0 = 1/k – 0.0103
k = 97.0 when q = qmax
Because k, at qmax, is approaching kj, this model is valid for high density
conditions only.
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