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In calculus and differential equations, a standard example of word problems are mixing problems, with some number of tanks, and brine often being an output of the system. With one tank, I can imagine some relation to real world scenarios, as people actually make brine, or maintaining aquariums (perhaps not varying salt content, but doing something like controlling pH).

What I have more trouble motivating with good concrete practical applications is systems where multiple tanks are involved and the tanks flow into each other at possibly different rates, though no doubt there are many examples from environmental sciences, chemistry, etc. Can someone provide convincingly practical mixing examples involving multiple tanks (loosely interpreted)?

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It can be seen as a simplified model for demographics. You have several geographical areas, each with some input (births and immigration) and some output (death and emigration). Some of the migration goes between the given areas and some go to or come from the outside. Now say you want to know the number of people after a certain time with a given characteristic. With a few assumptions (people with and without that characteristic are equally likely to move / die / have children etc.) you get exactly the same dynamics.

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Chemical Engineering Application

Systems of mixing tanks flowing into the next in sequence are very typically applied in chemical engineering.

Due to increasing clean water supply issues around the world, a relevant example involves the use of microbes (micro-organisms) to purify industrial waste water.

Because the waste water is continuously produced as a by-product of some industrial process plant (e.g. a plastics production plant or mineral processing plant), the microbes and the waste water may be fed at specific rates in order to optimize the purification rate. While the industrial waste water flows from one tank to the next, each of (say) four tanks may have different microbial feed rates in order to carefully control the total rates of reaction necessary to achieve desired purity levels of the final clean water.

Later, the pure water can be separated from the microbes and sent for further processing before it can finally be portable enough for use.

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In biology, somewhat more complicated versions of these standard ODEs can be used to model the interaction of a cell with its surroundings. Here the cell is one tank, which is coupled only to the other tank, while the surroundings are the second tank, which are also coupled to an external input and drain (given by the "farther away surroundings", such as the kidneys). The actual dynamics of the transport are more complicated because of the effects of the cell membrane, but the mass balance principles are identical.

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