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Case 3: r1= r2=r, k=k1>0, V=V0
Now we have
This is separable. We get
Integrating both sides gives us
or
And thus
This gives us
To find C:
And thus
Finally:
We can see that case 1 is just a special case of case 3.
An example
Say we have V=V0=2000 l, r1 = r2 = r = 20 l/s, k=k1=10/2000 kg/l and y0 = 50 kg. This would give us the solution
This can be seen as the green curve in the figure. We can see that we asymptotically get toward the concentration of the inflow.
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