Mixing and tank problems use calculus to track how the amount of a substance changes in a liquid over time. These problems appear in chemistry, environmental science, medicine, and engineering whenever material flows into and out of a container. The main idea is to model accumulation using an input rate and an output rate.
A differential equation then predicts the concentration at any time.
Key Facts
- Basic model: dA/dt = rate in - rate out
- Amount of solute: A(t) = concentration times volume
- Rate in = inflow concentration times inflow rate
- Rate out = tank concentration times outflow rate = A(t)/V(t) times outflow rate
- Volume changes by dV/dt = inflow rate - outflow rate
- If volume is constant, many mixing models have the form dA/dt + kA = c
Vocabulary
- Solute
- The substance being dissolved or mixed in the liquid, such as salt in water.
- Concentration
- The amount of solute per unit volume of solution, often measured in grams per liter.
- Flow rate
- The volume of liquid entering or leaving the tank per unit time.
- Rate in
- The amount of solute entering the tank per unit time.
- Rate out
- The amount of solute leaving the tank per unit time, based on the current tank concentration.
Common Mistakes to Avoid
- Using the inflow concentration for the outflow concentration is wrong because the liquid leaving the tank has the same concentration as the well mixed tank, not necessarily the incoming liquid.
- Forgetting that volume can change is wrong because unequal inflow and outflow rates make V(t) depend on time, which changes the rate-out term.
- Writing rate out as just the outflow rate is wrong because rate out must measure solute per time, so it needs concentration times volume flow rate.
- Solving for concentration before solving for amount is often wrong because the differential equation is usually simplest in terms of A(t), the amount of solute.
Practice Questions
- 1 A 100 L tank contains 20 g of salt. Brine with concentration 3 g/L flows in at 4 L/min, and the well mixed solution flows out at 4 L/min. Write the differential equation for A(t), the amount of salt in grams.
- 2 A 50 L tank initially contains pure water. Saltwater with concentration 2 g/L flows in at 3 L/min, and the mixture flows out at 3 L/min. Find the limiting concentration as t becomes very large.
- 3 A tank has inflow greater than outflow, so its volume increases over time. Explain how this changes the rate-out term compared with a constant-volume tank.