# Solving Dilution Problem with Water Level Change & Brine Addition

• tickle_monste
In summary, the author illustrates the methods for solving the problem when adding more brine to the tank and when the water level is changing, but does not provide a method for solving the problem when the water level is changing and the brine is not being added.
tickle_monste
Ok, I'm working through some dilution/accretion problems in my ODEs book (not homework), all of them centered around draining brine from a tank. The author illustrates the method for solving the problem when you're adding more brine to the tank, and the total water level stays the same, you simply separate the variables. The author also illustrates the method for solving the problem when the water level is changing, but you're not adding more brine to the tank. Again you simply separate the variables.

And that's it, that's all my book explains. It doesn't explain at all what to do when you're both changing the water level and adding more brine to the tank. The author doesn't even hint at a method for solving it:

Adding brine, but not changing the water level:

100 gallon tank, 3 gals/min brine flow in at 2 lbs of salt per gallon, 3 gals/min of the mix flow out.
dx = 6dt (brine) - (x/100)3dt.
dx/(x-200) = -.03dt.
Separable, hooray.

Not adding brine, changing the water level:

100 gallon tank, 2 gals/min fresh water flow in, 3 gals/min of the mix flow out.
dx = (x/(100-t))3dt.
dx/x = (3/100-t)dt
Again, separable.

Adding brine, changing the water level:

100 gallon tank, 3 gals/min brine flow in at 2lbs of salt per gallon, 2 gals/min of the mix flow out.
dx = 6dt - (x/(100+t))2dt
That doesn't look very separable to me...

You are right, it isn't. However, it can be solved. First define $$\tau=t+100$$. Then:
$$x'=6-2x/\tau$$
which is homogeneous (invariant under $$x\rightarrow ax, \tau \rightarrow a\au$$). For that kind, you have the standart change $$z=x/\tau$$, which transforms it into the new equation
$$z'\tau=6-3z$$
which is now separable. The full solution is then
$$x=2(t+100)-\frac{C}{(t+100)^{2}}$$

Last edited:
I see how that works. Thanks a lot for the help

Actually on second thought I'm not quite sure what you're trying to specify with z'. What exactly are you differentiating Z with respect to?

He is differentiating the new dependent variable, z, with respect to the new independent variable, $\tau$.

that is $$x=z(\tau)\tau\rightarrow x'(\tau)=z'\tau+z$$

at what speed should a clock be moved so that it may appear to lose 1 minute in each hour?

## 1. How do you calculate the final concentration after adding water and brine to a solution?

To calculate the final concentration, you can use the following formula: Cf = (Ci x Vi + Cb x Vb) / (Vi + Vb), where Cf is the final concentration, Ci is the initial concentration, Vi is the initial volume, Cb is the concentration of the added brine, and Vb is the volume of the added brine.

## 2. What is the purpose of adding brine to a solution during dilution?

The purpose of adding brine is to maintain the original concentration of the solution while increasing its volume. This is commonly done in chemistry experiments or industrial processes to create a larger volume of a solution with a specific concentration.

## 3. How does the water level change when adding brine to a solution?

When brine is added to a solution, the water level will increase due to the added volume of the brine. This is a key factor in calculating the final concentration of the solution after dilution.

## 4. Can you use this method to dilute any type of solution?

Yes, this method can be used to dilute any type of solution as long as the added brine does not react with the solute in the solution. It is important to choose a suitable brine that will not affect the final concentration or any other properties of the solution.

## 5. How does temperature affect the accuracy of this dilution method?

Temperature can affect the accuracy of this dilution method as it can change the volume of the solution. To ensure accuracy, it is important to measure the volume of the solution and the added brine at the same temperature. If the temperatures are different, you can use temperature correction factors to adjust the volumes accordingly.

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