Solved Examples: Pollutant Concentration in Lake Systems using Differential Equations

📚 Quick Study Guide

• 🌊 Pollutant Input: The rate at which a pollutant enters the lake, often denoted by $P$ (mass/time).
• 💧 Flow Rate: The rate at which water flows into and out of the lake, denoted by $Q$ (volume/time).
• 📏 Volume: The volume of the lake, denoted by $V$ (volume). Assume constant volume for simplicity.
• 🧪 Concentration: The concentration of the pollutant in the lake, denoted by $C(t)$ (mass/volume) at time $t$.
• ➗ Differential Equation: The basic differential equation governing pollutant concentration is: $\frac{dC}{dt} = \frac{P}{V} - \frac{Q}{V}C$ This equation represents the change in concentration over time as a balance between pollutant input and outflow.
• ⏱️ Equilibrium Concentration: The steady-state concentration $C_{eq}$ is reached when $\frac{dC}{dt} = 0$. Solving for $C_{eq}$ gives: $C_{eq} = \frac{P}{Q}$
• 📈 Solving the DE: The solution to the differential equation often takes the form: $C(t) = C_{eq} + (C_0 - C_{eq})e^{-\frac{Q}{V}t}$ where $C_0$ is the initial concentration at $t=0$.

Practice Quiz

1. What does the term $\frac{P}{V}$ represent in the differential equation for pollutant concentration in a lake?
   1. The rate of pollutant outflow.
   2. The rate of pollutant input per unit volume.
   3. The total amount of pollutant in the lake.
   4. The rate of water inflow.
2. What is the equilibrium concentration ($C_{eq}$) of a pollutant in a lake when the input rate is $P$ and the outflow rate is $Q$?
   1. $P \times Q$
   2. $\frac{Q}{P}$
   3. $\frac{P}{Q}$
   4. $P + Q$
3. A lake has a volume of $10^6$ $m^3$ and an inflow/outflow rate of $10^4$ $m^3$/day. A pollutant enters at a rate of 10 kg/day. What is the equilibrium concentration of the pollutant in the lake?
   1. 0.001 kg/$m^3$
   2. 0.01 kg/$m^3$
   3. 1 kg/$m^3$
   4. 0.1 kg/$m^3$
4. What does the term $e^{-\frac{Q}{V}t}$ represent in the solution of the differential equation?
   1. The rate of pollutant input.
   2. The exponential decay of the initial concentration difference from equilibrium.
   3. The total concentration at time $t$.
   4. The rate of water outflow.
5. A lake with a volume of $V$ has an inflow/outflow rate of $Q$. What is the time constant for the system to reach equilibrium?
   1. $V \times Q$
   2. $\frac{Q}{V}$
   3. $\frac{V}{Q}$
   4. $\frac{1}{VQ}$
6. Initially, a lake is pollutant-free ($C_0 = 0$). If a pollutant starts entering at a constant rate, how does the concentration change over time?
   1. It decreases exponentially.
   2. It increases linearly.
   3. It increases exponentially, approaching the equilibrium concentration.
   4. It remains constant.
7. If the inflow rate $Q$ increases, how does the equilibrium concentration $C_{eq}$ change, assuming the pollutant input rate $P$ remains constant?
   1. It increases.
   2. It decreases.
   3. It remains the same.
   4. It oscillates.