23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
a. Write an initial value problem for the mass of the substance.


A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

Verified step by step guidance

1. Define the variable \(m(t)\) as the mass of the pollutant in the pond at time \(t\) (in hours). Since the pond volume is constant at 1,000,000 liters, the concentration at time \(t\) is \(c(t) = \frac{m(t)}{1,000,000}\) grams per liter.
2. Set up the initial condition: at time \(t=0\), the mass of pollutant is \(m(0) = 20 \text{ g/L} \times 1,000,000 \text{ L} = 20,000,000\) grams.
3. Write the rate of change of mass \(\frac{dm}{dt}\) considering the inflow and outflow. Since pure water flows in, no pollutant enters, so the inflow term is zero. The outflow removes pollutant at a rate proportional to the concentration and the outflow rate: \(\frac{dm}{dt} = - (\text{outflow rate}) \times c(t) = -1200 \times \frac{m(t)}{1,000,000}\).
4. Formulate the initial value problem (IVP) as the differential equation: \(\frac{dm}{dt} = -\frac{1200}{1,000,000} m(t)\) with initial condition \(m(0) = 20,000,000\).
5. To find the time \(t\) when the concentration is 10% of the initial value, set \(c(t) = 0.1 \times 20 = 2\) g/L, which means \(m(t) = 2 \times 1,000,000 = 2,000,000\) grams. Solve the IVP for \(m(t)\) and then solve for \(t\) when \(m(t) = 2,000,000\).

Key Concepts

Formulating Initial Value Problems (IVPs)

An initial value problem involves setting up a differential equation that models the rate of change of a quantity along with an initial condition. In this context, it means expressing how the mass of the pollutant changes over time based on inflow, outflow, and mixing, starting from the known initial mass.

Mixing and Flow in Stirred Tank Models

A stirred tank model assumes the contents are perfectly mixed, so the concentration is uniform throughout. The inflow and outflow rates affect the concentration by diluting or removing the substance, and the volume remains constant if inflow equals outflow, simplifying the mass balance.

Solving First-Order Linear Differential Equations

The mass or concentration change in the pond can be described by a first-order linear differential equation. Solving this equation involves integrating to find the concentration as a function of time, which allows determining how long it takes for the concentration to reach a specified fraction of its initial value.