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.
Understanding Calculus: Mixing and Tank Problems
A tank model depends on a hidden assumption called perfect mixing. It means that as soon as liquid enters, the dissolved material spreads evenly through the whole tank. The liquid leaving through a drain then has the same concentration as the liquid inside.
This is often reasonable when the tank is stirred or when incoming flow creates strong turbulence. It can fail in a large lake, a slow moving pipe, or a container with layers of different density.
In those cases, salt, dye, or pollution may remain concentrated in one region. A simple tank equation is useful because it replaces a complicated fluid motion with one average amount.
The volume deserves careful attention. When liquid enters faster than it leaves, the tank becomes fuller, so the concentration in the drain must be based on an increasing volume. When more leaves than enters, the volume shrinks and a small amount of solute can become more concentrated.
Students should find the volume function before building the equation for the dissolved substance. They should then check the time interval.
A draining tank can reach zero volume, and a filling tank can overflow. A formula may still produce numbers after that point, but those numbers no longer describe the real container.
For a constant-volume tank, the equation often has a steady concentration. This is the concentration reached after a long time if the incoming conditions stay unchanged. At first, the amount may change quickly because the tank differs greatly from the incoming liquid.
Later, the change slows as material leaving the tank nearly balances material entering it. The solution usually contains an exponential term that gets smaller over time. Its rate of decrease depends on how quickly liquid is replaced.
A high flow rate refreshes the tank sooner. The initial amount affects the early behavior, but it does not affect the final steady concentration when flow continues indefinitely.
These models appear in water treatment systems, where chemicals are added at controlled rates. They help describe medicine entering and leaving the bloodstream, although real biological systems can need more detailed models. Environmental scientists use related ideas for pollutants in reservoirs.
Unit checks are one of the best ways to catch mistakes. A concentration might be measured in grams per liter, while flow is liters per minute, giving grams per minute for a solute rate. Keep time units consistent throughout.
It also helps to state what the unknown represents. Use amount for the main differential equation, then divide by volume only when concentration is needed. This prevents a common error of treating concentration as if it were the total mass of dissolved material.
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.