I have a variation on the concentrating tank problem that I'm having a bit of trouble solving. I have a tank of 10 kg of pure water at time 0. I add a time dependent concentration of salt and remove the same volume of pure water so that the tank volume never changes. Once the tank has 1 kg of salt, I stop the problem. So unlike the typical problem where I have a known initial concentration, I instead have a target final concentration, but the time of that final concentration is unknown in that it depends on the flow rate.(adsbygoogle = window.adsbygoogle || []).push({});

$$ \frac{dm_{salt}}{dt} = in - out $$

There's no salt exiting, so it becomes,

$$ \frac{dm_{salt}}{dt} = \dot{m}_{in} \frac{m_{salt}(t)}{m_{tank}} $$

Separate the variables and integrate to get

$$ln(m_{salt}) = \frac{\dot{m}_{in} t}{m_{tank}} + C$$

$$m_{salt}(t) = Ae^{\frac{\dot{m}_{in} t}{m_{tank}}}$$

Now I need an initial condition to get the particular solution, but I can't use

$$m_{salt}(0) = 0$$

and I'm not sure how to use my knowledge of the final concentration, since I don't know what time it will occur.

Your help is appreciated.

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# Concentrating Salt Tank

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