Step2: Gather Information

This problem seems, at first, to be rather complicated because of the number of parameters, functions, and equations that are involved. You will see, however, that the problem is not too difficult if the information is carefully organized.

The initial value problems for the remaining BOD, , and the DO deficit, , form the foundation of the problem.

> eqnL := diff( L(t), t ) = -kd*L(t);

> icL := L(0) = BODu;

> eqnD := diff( DD(t), t ) = kd*L(t) - kr*DD(t);

> icD := DD(0) = Do;

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The parameters in this model are the deoxygenation and reaeration rates and and the initial conditions are the ultimate BOD, , and the oxygen deficit level, , when the pollutant is added to the water.

The solutions to these initial value problems are, provided :

> SOLN := dsolve( { eqnL, eqnD, icL, icD }, { L(t), DD(t) } ):

> solnD := DD = eval( DD(t), SOLN ): solnD;

> solnL := L = eval( L(t), SOLN ): solnL;

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Other quantities of interest in the analysis include the amount of oxygen consumed by the waste organisms,

> solnY := y = L[0] - rhs( solnL );

and the dissolved oxygen level in the stream,

> solnDO := DO = DO[sat] - rhs(solnD);

where is the temperature-dependent saturated DO level (which is obtained from the table). Also, let (C) denote the water temperature and (km/day) the stream velocity. Specific values for each of the parameters are collected below:

> DOvals := [ kd=0.4, kr=2.0, DOo=2.2, BODu=54.8 ];

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