Compartmental Analysis - Page 4

Applications of Differential Equations

Compartmental Analysis

Solving The Example in Mathematica

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Mathematica should have given you a graph which looked a lot like this:

Obviously we're getting the exponential decay that we expected. It looks like the solution is decaying towards an asymptote of approximately y = 800, and is getting pretty close to this asymptote by the time 400 hours have passed. Can you explain, simply using a common sense argument, why the amount of salt in the tank approaches 800 pounds in the long haul?

The argument could go something like this.  The more time that passes, the less of the original salt solution is left in the tank. Over an extended period, the fluid in the tank will almost entirely consist of what has flowed in through the inflow pipe. This incoming liquid contains 2 pounds of salt per gallon, so since the tank holds 400 gallons of fluid, if each gallon carries 2 pounds of salt there should be a total of 800 pounds of salt in the tank.

For fun, let's have Mathematica take the limit of mix1 as t→∞. Issue the command:

Limit[mix1,t->Infinity]

(By the way, if you don't want to bother typing Infinity out in full, option-5 on the Macintosh keyboard will type the ∞ symbol for you.) Hopefully you got exactly the answer that your common sense told you to expect.

Moving on...

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