3.2 Compartmental Analysis

Mixture Problems

In these mixture problems a substance dissolved in a liquid is entering and exiting a holding tank. The

differential equation modeling this situation is:

𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑥(𝑡) = 𝑅𝑎𝑡𝑒 𝑥(𝑡) 𝑒𝑛𝑡𝑒𝑟𝑠 − 𝑅𝑎𝑡𝑒 𝑥(𝑡) 𝑒𝑥𝑖𝑡𝑠

𝑑𝑥

𝑑𝑡= 𝑟𝑖𝑛 − 𝑟𝑜𝑢𝑡

where,

𝑥(𝑡) = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑑𝑖𝑠𝑠𝑜𝑙𝑣𝑒𝑑 𝑖𝑛 𝑎 𝑙𝑖𝑞𝑢𝑖𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡

𝑟𝑖𝑛 = (𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔) × (𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝑙𝑖𝑞𝑢𝑖𝑑 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔)

𝑟𝑜𝑢𝑡 = (𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 𝑒𝑥𝑖𝑡𝑖𝑛𝑔) × (𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝑙𝑖𝑞𝑢𝑖𝑑 𝑒𝑥𝑖𝑡𝑖𝑛𝑔)

Examples

1. Water flows into a pond at a rate of 10 𝑚3 𝑚𝑖𝑛⁄ and out at the same rate. The volume of the pond is 1000 𝑚3.

Initially, the pond contains 100 grams of pollutants. The concentration of pollutants in the water flowing into the

pond is 2 𝑔 𝑚3⁄ . Assuming immediate and uniform dispersion of the pollutants, find a formula for the number

of grams of pollutants in the pond, 𝑥 = 𝑥(𝑡).

𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝑙𝑖𝑞𝑢𝑖𝑑 𝑒𝑥𝑖𝑡𝑖𝑛𝑔 = 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎𝑡 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒, 𝑡

𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑛𝑘 𝑎𝑡 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒, 𝑡

2. The air in a small room 12 ft by 8 ft by 8 ft is 3% carbon monoxide. Starting at 𝑡 = 0, fresh air containing no

carbon monoxide is blown into the room at a rate of 100 𝑓𝑡3 𝑚𝑖𝑛⁄ . If air in the room flows out through a vent

at the same rate, when will the air in the room be 0.01% carbon monoxide?

Population Models

𝑑𝑝

𝑑𝑡= 𝑘𝑝, 𝑝(0) = 𝑝0

Where k is a constant of proportionality

Note: 𝑘 > 0 for growth and 𝑘 < 0 for decay

3. Solve the differential equation for population growth.

𝑑𝑝

𝑑𝑡= 𝑘𝑝, 𝑝(0) = 𝑝0

This is the Malthusian or exponential, law of population growth and it assumes that members of a population only die

by natural causes. This means it is not very accurate for a large population or over a long period of time.

The Logistic Equation

The logistic equation has proven to be more accurate in predicting population growth because it takes into account

competition within a population.

𝑑𝑝

𝑑𝑡= −𝐴𝑝(𝑝 − 𝑝1) , 𝑝(0) = 𝑝0

where a and b are constants.

Solving this equation gives us the logistic function

𝑝(𝑡) =𝑝0𝑝1

𝑝0 + (𝑝1 − 𝑝0)𝑒−𝐴𝑝1𝑡

4. In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006

the population had grown to an estimated 6000.

a. Use the Malthusian law for population growth to estimate the alligator population on the Kennedy

Space Center grounds in the year 2020.

b. Suppose we have the additional information that the population of alligators on the grounds of the

Kennedy Space Center grounds in 1993 was estimated to be 4100. Use the logistic model to estimate the

population of alligators in the year 2020. What is the predicted limiting population?

𝑝1 = [𝑝𝑎𝑝𝑏 − 2𝑝0𝑝𝑏 + 𝑝0𝑝𝑎

𝑝𝑎2 − 𝑝0𝑝𝑏

] 𝑝𝑎

𝐴 =1

𝑝1𝑡𝑎ln [

𝑝𝑏(𝑝𝑎 − 𝑝0)

𝑝0(𝑝𝑏 − 𝑝𝑎)]

5. If initially there are 300g of a radioactive substance and after 5 years there are 200g remaining, how much time

must elapse before only 10g remain?