Mixing Salt & Water with Separable Differential Equations

in #steemiteducation6 years ago (edited)

Let's do some more mathematical modelling. Differential equations can be helpful in calculating the concentration of a mixture at any given time in a reservoir.

Mixing.png
Image source

For instance, let's say we have a tank which is initially filled with 15kg of salt dissolved in 3000L of water. Brine, at a concentration of 20 grams per litre (0.02kg/L), is flowing into the tank at a rate of 10 litres per minute. It is thoroughly mixed with the solution that's already in the tank. At the bottom of the tank, there is a pipe which is draining the mixed solution at the same flow rate as the brine flowing in. How much salt is in the tank after 1 hour?

To solve this problem, we let y(t) be the amount of salt in the tank at any given time, t. At any time, the rate of change of salt dy/dt is the difference between the rate salt coming into the tank to the rate of salt leaving the tank. That is...

d1.png

Now the rate of salt coming into the tank is simply the concentration of the brine by the flow rate of the solution. That is...

d2.png

The rate of salt leaving the tank, after being thoroughly mixed is similarly, the concentration of the solution by its flow rate...

d3.png

Ok, disregarding the units, we have a differential equation depicting the rate of change of salt in our tank, which we can solve...

d4.png

Note that the above is a separable, first-order, linear differential equation. Now to write it in differential form...

d5.png

...and solve by integrating both sides...

d6.png

Now that we have the general solution, let's apply the initial condition where we start with 15 kg in the 3000 L tank. So, at time 0...

d7.png

And thus the particular solution for this initial value problem is...

d8.png

Now, to find the amount of salt mixed in the tank after 1 hour, at time t = 60min...

d9.png

A graph of the solution is shown below...

MixingGraph.pngCreated with: www.desmos.com/calculator

As we can see, the amount of salt in the tank at the start is 15kg. As time goes on, the amount of the salt approaches a maximum of 60kg.

That will complete this tutorial. Below is a list of tutorials I've created so far on Differential Equations:

  1. Introduction to Differential Equations - Part 1

  2. Differential Equations: Order and Linearity

  3. First-Order Differential Equations with Separable Variables - Example 1

  4. Separable Differential Equations - Example 2

  5. Modelling Exponential Growth of Bacteria with dy/dx = ky

  6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky

  7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky

  8. Mixing Salt & Water with Separable Differential Equations

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