2. A 100 L mixing tank initially contains 75 L of brine solution with 300 g of dissolved salt. Brine solution with a concentration of 12 g/L flows into the tank at a rate of 5 L/min. The solution inside the tank is kept well- stirred and is discharged at the same rate. Determine the a) concentration of brine inside the tank after 10 minutes, and b)time it takes to achieve a concentration of 10 g/L.
application of differential equations salt solution
Given,
- The volumetric flow rate at the entrance,
- The volumetric flow rate at the end,
- The concentration of the substance at the entrance,
- The initial volume of solution at ,
Since the given problem involves mix of non reactive fluids, the equation will be as follows:
Rate of change of substance in a volumeRate of entranceRate of exit
Since is not given, equation 1 can be rewritten as,
where
Hence,
Substitute the given values in the equation above.
Equation 1 is a separable differential equation.
From 1,
To find , use the fact that at time , .
Therefore,
Therefore,
Therefore, the concentration of brine inside the tank after minutes is given by :
Therefore the concentration of brine inside the tank after 10 minutes is,
Observe that the since the rate of input and the rate of discharge is the same the mixing tank contains L of brine solution even after 10 minutes.
Therefore, the concentration of brine inside the tank after 10 minutes is,
or equivalently
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