Page 215 - Soil and water contamination, 2nd edition

202 Soil and Water Contamination

Box 11.IV (continued) Physics of surface water flow

the case in rivers. Using Q = u ⋅ A, Equations (11.IVa) and (11.IVb) can be solved for Q.
Because these Saint-Venant equations are not amenable to analytical solutions except in a
few simple cases, the equations are usually solved numerically.

Example 11.3 Advection

-1
In a river with a mean flow velocity of 0.80 m s , the chloride concentration increases
due to increasing wastewater discharge from an upstream salt mine at location x = 0 km.
-1
-1
At x = 4 km, the chloride concentration increases linearly from 20 mg l to 140 mg l
between 4 a.m. (04.00 h) until 4 p.m. (16.00 h). Calculate the chloride concentration at
x = 40 km and 8 p.m. on the same day.
Solution
-1
-1
First, convert the flow velocity from m s to km h :
-1
-1
-1
ū = 0.80 m s = 0.80 m s × 3600 s h = 2.880 km h -1
x
Next, assume that t is the time that the chloride concentration at location x = 4 km is
0
equal to the concentration at x = 40 km and 8 p.m. Thus, the initial condition at x = 4
km and t = t is:
0
C , 4 ( t ) C ) 4 (
0 0
Use equation (11.15) to define the concentration at x = 40 km at t = t (8 p.m.):
C ( 40 ,t ) C 0 ( 40 . 2 880 (t t 0 ))
Because we have assumed that C(4,t ) = C(40,t), we may write:
0
C 0 ) 4 ( C 0 ( 40 . 2 880 (t t 0 ))
4 40 . 2 880 (t t 0 )

. 2 880 (t t ) 36
0
t t 12 5 . h
0
Thus, it takes 12.5 hours for the water to travel from location x = 4 km to location x = 40
km. This can also be calculated directly from Δt = Δx/ū . From the calculation it appears
x
that t is 12.5 hours before 8 p.m. = 7.30 a.m., which is 3.5 hours after the chloride
0
concentration started increasing. At location x = 4 km, the chloride concentration
-1
-1
-1
-1
increases from 20 mg l to 140 mg l in 12 hours, so 10 mg l h . This implies that
-1
-1
-1
after 3.5 hours, the concentration had increased by 10 mg l h × 3.5 h = 35 mg l . The
chloride concentration at location x = 0 and t = t is
0
-1
C , 4 ( t ) 20 35 55 mg l .
0
This is also the concentration at x = 40 km and 8 p.m.

10/1/2013 6:44:51 PM
Soil and Water.indd 214 10/1/2013 6:44:51 PM
Soil and Water.indd 214