FUNDAMENTALS OF FATE AND TRANSPORT - Water Quality Engineering in Natural Systems - page 70

EXAMPLE 3.24

water surface

D y

The velocity and depth are measured at 1-m intervals
across a 10-m-wide stream and the measurements are
as follows:

h i

Distance velocity Depth

bottom elevation

(m) (m/s) (m)

0 0.00 0.0

Figure 3.26. Discretization of channel cross section.

1 0.15 1.2

2 0.27 2.6

3 0.32 3.5

y u h Δ A Δ Q u′

4 0.38 3.8

(m) (m/s) (m) (m 2 ) (m 3 /s) (m/s)

5 0.40 4.0

0 0.00 0.0 0.0 0.000 −0.33

6 0.40 3.6

1 0.15 1.2 1.2 0.180 −0.18

7 0.37 3.5

2 0.27 2.6 2.6 0.702

−0.06

8 0.30 2.7

3 0.32 3.5 3.5 1.120 −0.01

9 0.15 1.7

4 0.38 3.8 3.8 1.444 0.05

10 0.00 0.0

5 0.40 4.0 4.0 1.600 0.07

6 0.40 3.6 3.6 1.440 0.07

7 0.37 3.5 3.5 1.295 0.04

The transverse diffusion coefficient in the stream is esti-
mated as 200 cm 2 /s; estimate the longitudinal dispersion
coefficient.

8 0.30 2.7 2.7 0.810 −0.03

9 0.15 1.7 1.7 0.255

−0.18

10 0.00 0.0 0.0 0.000 −0.33

Total: 26.6 8.846

Solution

Based o n these results, A = 26.6 m 2 , Q = 8.846 m 3 /s, and
hence u =

From the given data: Δ y = 1 m, W = 10 m, ε y =
200 cm 2 /s = 0.02 m 2 /s, and u ( y ) and h ( y ) are given. The
total area, A , total flow, Q , and average velocity,
u , in the stream are given by

8 846 26 6 . / . = 0 333 . m/s . The value of u ′ is

based on the definition of u u u u 0.333 m/s .

Since the depth varies across the channel, the longitu-
dinal dispersion coefficient, K x , is given by Equation
(3.243) as

′ = − = −

10 10

∑ ∑

A = ∆ A = h y ∆

i i

i = 1 i = 1

10 10

1 1

W y y

∑ ∑

∫ ∫ ∫

′ ( ) ′ ( )

Q = ∆ Q = u h y ∆ K = − u h y u h y dy dy dy

i i i

x

( )

A ε h y

0 0 0

y

i = 1 i = 1

I

1

Q
A

I

u =

2

I

3

(3.244)

where h i and u i are the depth and velocity at the i th
measurement location. The discretization of the channel
cross section is shown in Figure 3.26.
The computations to calculate A , Q , and u are sum-
marized in the following table:

where the subintegrals used in evaluating K x are denoted
by I 1 , I 2 , and I 3 . using Δ y = 1 m, numerical evaluation of
the integral for calculating K x using the above-tabulated
data is summarized in the following table: