Help with this problem?

Question

A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min
(a) How much salt is in the tank after t minutes?(kg)
(b) How much salt is in the tank after 20 minutes?(kg)

Answer 1

Maybe you should look at the example just like it in your differential equations textbook.

Answer 2

This problem will lead to a 1st order non homogeneous differential equation.

If S is the mass of salt in your tank at any given time, then

dS/dt = 0.05*5 + 0.04*10 -S*15/1000.

since in a dt (of 1min), you get 0.05kg/L * 5L + 0.04kg/L * 10L in, and a quantity of S/1000 kg/L * 15L of salt out of your tank.
(you need to make sure that there is 1000L of liquid at all times in the container... which is the case, or the equation you have to solve would be a bit more complicated)

After that, you only need to follow the instructions given in the following link to solve the problem completely.

Once you have solved that equation, you can answer the 2 questions easily

have fun.


PS : seeing the equation, i can already tell you the solution is has a (negative_constant) * exp( negative_constant_bis * t ) in it. If you don't have it, redo your maths.

Answer 3

I am not going to think about this, but I can tell you that the salt concentration does not remain constant. You start off with pure water, you introduce salt water, mix it, then evacuate the same quantity that you introduced (but this now is from the whole tank) - so the salt inside grew. This happens every second so you need differential equations.
So Finndo must be right. Besides he (or she) is a scientist.
I am an engineer.

Answer 4

Incoming supply of Brine:
5L/min + 5(0∙05)kg of salt.
+10L/min +10(0∙04)kg of salt.
--------------------------------------
15L/min + 0∙45kg of salt.

Flow of water in = Flow of water out.
=> The salt concentration remains constants.

(a) After t minutes = 0∙45kg.
(b) After 20 minutes = 0∙45kg.