For anything that accumulates, the rate of change is what comes in minus what goes out. That single sentence builds most of these models.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Rate in
Multiply the inflow rate by the incoming concentration. This is usually a constant.
Rate out
Multiply the outflow rate by the current concentration, which is the amount present divided by the volume.
Why it is linear
The outflow term is proportional to the amount present, so the equation lands in exactly the form of the previous lesson.
Steady state
Setting the rate of change to zero gives the long-run amount. The tank eventually matches the incoming concentration.
Circuits use the same equation
A resistor and capacitor charging toward a supply voltage obey the identical mathematics, with charge in place of salt.
Rate in minus rate out
The change in the amount of substance equals what enters minus what leaves. Writing that sentence as an equation is the whole modelling step, and the rest is technique.
The outflow depends on the current amount
Concentration is amount over volume, so the rate out involves the unknown function. That dependence is what makes the problem a differential equation rather than arithmetic.
Watch whether the volume changes
If inflow and outflow rates differ, the volume is a function of time and appears in the concentration. Assuming constant volume when it is not is the standard modelling error here.
Circuits have the same structure
An RL or RC circuit produces a linear first-order equation identical in form to a mixing problem. The same mathematics describing brine tanks describes current in a coil, which is a genuine unification.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.03)
Step 3: Watch an Example
One step at a time.
Watch Dmitri Build a Tank Model
Dmitri has a 200 litre tank with brine at 3 grams per litre entering at 5 litres per minute, and the well-mixed solution leaving at the same rate.
- Step 1
He computes the rate in: 5 litres per minute times 3 grams per litre, which is 15 grams per minute.
Step 4: Your Turn
Practice makes it stick.
The Inflow
Problem 1 of 2
Brine at 4 grams per litre entering at 6 litres per minute. What is the rate in, in grams per minute?
The Steady State
Problem 2 of 2
dS/dt = 15 − S/40. What is the steady-state amount of salt, in grams?
In Minus Out
1 of 8
2 grams per litre entering at 10 litres per minute. What is the rate in?
2 of 8
dS/dt = 20 − S/50. What is the steady-state amount?
3 of 8
A 300 litre tank holding 900 grams of salt. What is the concentration in grams per litre?
4 of 8
Inflow 4 litres per minute and outflow 4 litres per minute. Does the volume change? 1 yes, 0 no.
5 of 8
dS/dt = 12 − S/25. What is the steady-state amount?
6 of 8
At the steady state, what is dS/dt?
7 of 8
Match each part of the model to what it represents.
Tap a card on the left to start.
8 of 8
A 100 litre tank of pure water, so no salt at all. What is the initial concentration?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dS/dt = 8 − S/30. What is the steady-state amount?
Question 2 of 2
How is a mixing problem set up?
What You Learned
- For anything that accumulates, the rate of change is the rate in minus the rate out.
- The outflow depends on the current concentration, which makes the equation linear.
- Setting the rate to zero gives the steady state the system approaches.