Under what flow conditions is the above model valid?

Question

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?

Answer 1

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.