Mixing Problems

Differential Equations · Axiom Academy

Mixing Problems: The Salt Tank One tank, one rule — salt in minus salt out — turns into the first-order linear DE you already know how to solve. A 100-liter tank starts with 10 kg of dissolved salt. Fresh brine flows in at 5 L/min carrying 0.2 kg/L , mixes instantly, and the same 5 L/min flows back out — so the volume never changes, only the concentration does. Drag time forward and watch the tank fill with salt — fast at first, then slower and slower as it nears a ceiling. That curve IS the differential equation's solution. How long until the tank nearly saturates? Flip the question around: pick a salt amount you want to see, and read off the minute it happens. Same solution, rearranged for t . Where does the tank always end up? Change what's flowing IN, and watch where the tank eventually settles. With equal in/out flow rates, the tank always drifts toward the inflow concentration — no matter what you start with. One balance, one linear DE: . When the flow rates match, the volume stays fixed and the solution is a clean exponential settling toward S_ eq =c_ in V with time constant . The exact same balance — rate in minus rate out — models drug clearance in the bloodstream, pollutant levels in a lake, and reactant concentration in a chemical reactor.

This is the written version of the interactive lesson above. See the full Differential Equations course.