Step 3: Generate and evaluate potential solutions

To begin the analysis, the parameters can be used to graph the sag curve. An estimate of the sag time can be obtained from the graph of the DO sag curve. The graph will be created by finding a general expression for and then substituting all relevant parameter values.

The general solution to the Streeter-Phelps equation provides an explicit formula for the oxygen deficit :

> solnD;

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The corresponding equation for the DO level in the water is obtained from the fact that the total dissolved oxygen is conserved:

> DOconserv := DO + DD = DOsat;

Now, the formula for dissolved oxygen is found to be

> DOeqn := op( solve( DOconserv, {DO} ));

> DOeqn := eval( DOeqn, solnD );

> DOeqn1 := collect( DOeqn, { exp(-kd*t), exp(-kr*t) } );

Before the DO sag curve can be plotted, it is necessary to determine the initial oxygen deficit, , and the DO saturation, . The saturation level is, with =21C, =8.9 mg/L; the corresponding deficit is = = 8.9-2.2 = 6.7 mg/L:

> DOvals := [ op(DOvals), Do=6.7, DOsat=8.9 ];

> DOeqn2 := eval( DOeqn1, DOvals );

> plot( rhs(DOeqn2), t=0..14, title="DO sag curve" );

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The sag time is estimated, using Maple's graphics interface, to be =0.575 days, or 13.8 hours. Since the stream velocity is 20 km/day, the minimum DO level, which is (approximately) =0.224 mg/L, occurs 11.5 km downstream from the contamination injection point.

Observe that the initial DO level for this problem is almost 50 percent lower than the minimum level needed to support fish life. The DO level finally returns to a level of 4 mg/L after 2.525 days, when the spill is 50.5 km downstream. Beyond this point, the DO level increases but does not exceed the saturation level (8.9 mg/L).

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