Mixing Problems

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.

Conditions and key results

The model assumes instantaneous perfect mixing, known flow rates, conserved solute, and no reaction. If the volume varies, treating it as constant changes the coefficient and violates mass balance. The solution interval ends when the tank empties or another capacity condition is reached.

A reliable strategy

1. Define $A(t)$ and $V(t)$ with units and compute the volume function first.
2. Write each solute rate as flow times concentration, carefully using $A/V$ only for well-mixed outflow.
3. Form the linear or separable ODE $A'=\text{in}-\text{out}$ and solve with $A(0)$.
4. Report amount or concentration as asked, restrict the time interval, and test equilibrium or limiting behaviour.

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.

Common mistakes

Verification and reasonableness

• Check every term has units of amount per time.
• At equilibrium, verify input and output solute rates agree.
• Substitute $A(t)$ into the balance and confirm $A(0)$ and physical bounds $0\le A/V$.

Practice

1. What is the outflow salt rate for flow 4 L/min, volume 80 L, amount $A$ g?
2. If equal flows enter and leave, what happens to volume?
3. In the example, what is the limiting amount?

Further deduction

With unequal flows, suppose pure water enters at 2 L/min and mixture leaves at 3 L/min from an initial 100 L. Then $V=100-t$ and $A'=-3A/(100-t)$. Separation gives $A=C(100-t)^3$, and $A(0)=A_0$ yields $A=A_0[(100-t)/100]^3$ for $0\le t<100$. The domain endpoint is dictated by emptying, not by algebraic convenience.

A total-mass audit integrates the model: solute at time $t$ equals initial solute plus cumulative inflow minus cumulative outflow. Evaluating that identity from a numerical or symbolic solution checks more than a single substituted derivative and can reveal an incorrect volume function.

Related topics

• First-Order Linear Differential Equations
• Population Models
• Initial Value Problems