Mixing Salt & Water with Separable Differential Equations
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](https://steemitimages.com/DQmNcaUqiBwAimzfkc3gZu319mrjmooyyNtFDKsB5YuF2r8/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](https://steemitimages.com/640x0/https://steemitimages.com/DQmUcJNcqDhbYkbnQMbv5MyMPb5jeLoEY39SYp4NVorHLQg/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](https://steemitimages.com/DQmaCKLGhQkWz7jJhmtHzfY182SwPqrcLg2pDB2MqH4PP7o/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](https://steemitimages.com/640x0/https://steemitimages.com/DQmRHTbvgjzMXHeQbDaiZisyMUp1NHQimT6AAr78tbjiBaZ/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](https://steemitimages.com/DQmTjZ4g75RGBnGs5xh233MERCirsNWK3ZjXGchkWEd1ZWY/d4.png)
Note that the above is a separable, first-order, linear differential equation. Now to write it in differential form...
![d5.png](https://steemitimages.com/640x0/https://steemitimages.com/DQmac8Jvb7NL3VnChWJaWMRCDrCSYa1RxByrCekjzfyyNp2/d5.png)
...and solve by integrating both sides...
![d6.png](https://steemitimages.com/DQmb1p1N8nBu5E7Q33GpxcF5fv2qRRaHKSRRWSNYwRouGEm/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](https://steemitimages.com/640x0/https://steemitimages.com/DQmc1aY6u9UtAQzyDCYHge8rqwEPTrUKcWobWNwt6DaG7dj/d7.png)
And thus the particular solution for this initial value problem is...
![d8.png](https://steemitimages.com/DQmf4XVDfvWXkbhXv6Q7eRLohvACFenHkdcjspPeK15yYxS/d8.png)
Now, to find the amount of salt mixed in the tank after 1 hour, at time t = 60min...
![d9.png](https://steemitimages.com/640x0/https://steemitimages.com/DQmZExH5BDvfVwPBXPxy2dBd7mnXiXwmi7KD7eQFPZrHM5p/d9.png)
A graph of the solution is shown below...
![MixingGraph.png](https://steemitimages.com/DQmXmYd4sNTsiV9N1193rEXU54NCV67vB9bKcV6jZhsjJpY/MixingGraph.png)
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:
First-Order Differential Equations with Separable Variables - Example 1
Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky
Mixing Salt & Water with Separable Differential Equations
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