Mixture problems

Mixing problems occur constantly in the chemical industry, and a single bookkeeping idea handles them all.

Setup. A substance $S$ flows into a container of mixture at a constant rate, the mixture is kept uniform by stirring, and the uniform mixture simultaneously flows out at another (possibly different) rate (Fig. 10.2). Let $x(t)$ denote the amount of $S$ present at time $t$; if IN denotes the rate at which $S$ enters and OUT the rate at which it leaves, then

Here IN is (inflow concentration) $\times$ (inflow rate), while OUT is (current concentration, i.e. current amount / current volume) $\times$ (outflow rate) — so OUT typically depends on $x$ itself, making the resulting equation linear (rather than separable) in $x$.

Worked pattern (Example 10.30). A $1000$-litre tank holds $100$ g of dissolved salt. Brine enters at $10$ L/min, each litre carrying $5$ g of salt (so IN $=5\times10=50$ g/min); the well-stirred mixture leaves at the same $10$ L/min, so the outflow rate of salt is (fraction of tank drained per minute) $\times$ (current amount) $=\dfrac{10}{1000}x=0.01x$ g/min. This gives the linear equation

i.e. $\dfrac{dx}{x-5000}=-0.01\,dt$. Integrating: $x=5000+Ce^{-0.01t}$; the initial condition $x=100$ at $t=0$ fixes $C=-4900$, giving

as the amount of salt present at any time $t$. …