McKay, Donald. "Front matter" Multimedia Environmental Models ...

Worked Example 2.13
Consider a river into which the 1.8 million population of a city discharges a
detergent at a rate of 1 pound per capita per year, i.e., the discharge is 1.8 million
pounds per year. The aim is to calculate the concentrations at distances of 1 and 10
miles downstream from a knowledge of the degradation rate of the detergent and
the constant downstream flow conditions, which are given below. This can be done
in Eulerian or Lagrangian coordinate systems. The input data are first converted to
SI units.
The river flow rate is Uhw, i.e., 18270 m 3 /h. The rate constant k is 0.693/t 1/2, i.e.
0.096 h –1 . When the detergent mixes into the river, the concentration will be C O or
E/(Uhw) or 5.1 g/m 3 .
Lagrangian Solution
A parcel of water that maintains its integrity, i.e., it does not diffuse or disperse,
will decay according to the equation
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C = C O exp(–kt)
where t is the time from discharge. At 1 mile (1609 m), the time t will be 1609/U
or 1.47 h, and at 10 miles, it will be 14.7 h.
Substituting shows that, after 1 and 10 miles, the concentrations will be 4.4 and
1.24 g/m 3 . The chemical will reach half its input concentration when t is 0.693/k or
7.2 h, which corresponds to 7900 m or 4.92 miles. This Lagrangian solution is
straightforward, but it is valid only if conditions in the river remain constant and
negligible upstream-downstream dispersion occurs.
Eulerian Solution
Discharge rate 1.8 ¥ 106 lb per year 93300 g/h (E)
River flow velocity 1 ft/s 1097 m/h (U)
River depth 3 ft 0.91 m (h)
River width 20 yards 18.3 m (w)
Degradation half-life 0.3 days 7.2 h (t1/2) We now simulate the river as a series of connected reaches or segments or well
mixed lakes, each being L or 200 m long. Each reach thus has a volume V of Lhw
or 3330 m 3 . A steady-state mass balance on the first reach gives
input rate = UhwC O = output rate = UhwC 1 + VkC 1
where C O and C 1 are the input and output concentrations. C 1 is also the concentration
in the segment. It follows that