water plays in human society and the many competing interests Here we examine a

Water plays in human society and the many competing

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water plays in human society and the many competing interests Here we examine a very simplistic model for pollution of a lake The model illustrates some basic elements from which more complicated models can be built and analyzed Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (22/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake 2 Pollution in a Lake: Problem set up Consider the scenario of a new pollutant appearing upstream from a clean lake with volume V Assume that the inflowing river has a concentration of the new pollutant, p ( t ) Assume that the river flows into the lake at a rate, f ( t ) Assume that the lake is well-mixed and maintains a constant volume by having a river exiting the lake with the same flow rate, f ( t ), of the inflowing river Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (23/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake 3 Diagram for Lake Problem: Model with a linear first order DE for the concentration of the pollutant in the lake, c ( t ) f : flow rate f p : pollutant V : Volume c t ( ): concentration of pollutant in the lake Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (24/64)
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Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake 4 Differential Equation for Pollution in a Lake Set up a differential equation for the mass balance of the pollutant The change in amount of pollutant = Amount entering - Amount leaving The amount entering is the concentration of the pollutant, p ( t ), in the river times the flow rate of the river, f ( t ) The amount leaving has the same flow rate, f ( t ) Since the lake is assumed to be well-mixed, the concentration in the outflowing river will be equal to the concentration of the pollutant in the lake, c ( t ) The product f ( t ) c ( t ) gives the amount of pollutant leaving the lake per unit time Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (25/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake 5 Differential Equations for Amount and Concentration of Pollutant The change in amount of pollutant satisfies the model da ( t ) dt = f ( t ) p ( t ) - f ( t ) c ( t ) Since the lake maintains a constant volume V , then c ( t ) = a ( t ) /V , which also implies that c 0 ( t ) = a 0 ( t ) /V Dividing the above differential equation by the volume V , dc ( t ) dt = f ( t ) V ( p ( t ) - c ( t )) This is a Linear First Order DE If the lake is initially clean, then c (0) = 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (26/64) Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake 6 Solution of the DE:
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