Mixing Problems

A mass-balance method for well-stirred tank models, variable volume, units, and physically valid intervals.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

Let $A(t)$ be solute amount and $V(t)$ liquid volume in a well-stirred tank. The governing balance is $A'=\text{rate in}-\text{rate out}$. If inflow concentration is $c_{in}$ and flow $q_{in}$, input rate is $q_{in}c_{in}$. Under perfect mixing, outflow concentration is $A/V$, so output rate is $q_{out}A/V$.

Notation and mathematical language

Track amount units, such as grams, concentration in grams per litre, and flow in litres per minute. Volume satisfies $V(t)=V_0+(q_{in}-q_{out})t$ until overflow or emptying. The concentration $A/V$ is valid only while $V>0$.

Conceptual picture

The differential equation is conservation of mass. Perfect mixing makes tank concentration spatially uniform, so the exiting stream has the same concentration. Constant volume produces a linear equation with a stable equilibrium amount when inflow concentration is constant.

Fully worked example

Interpretation and application

Mixing equations model tanks, drug compartments, pollutants, and ventilation. Perfect mixing is an idealization; stratification or reaction can make observed concentration depart systematically from the one-compartment prediction.