Determining the Amount of Salt in a Brine Solution Over Time

Consider a tank initially containing 25 liters of brine solution with 2 kg of salt. Freshwater is added to the tank at a rate of 5 liters per minute, while the well-stirred solution exits the tank at a rate of 3 liters per minute. The question at hand is: what is the amount of salt present in the tank after 15 minutes?

Using numerical analysis, a small code was run to vary the step size, and the answer stabilized to approximately 1.53227 kilograms, indicating the remaining amount of salt.

A closer look at the mathematical model reveals an interesting dynamic. Since the freshwater inflow exceeds the solution outflow, the volume of solution in the tank is continuously growing. The volume of the solution in the tank can be described by the equation:

V 25 5t - 3t

where t represents the time in minutes and V the volume in liters. Simplifying this, we get:

V 25 2t

The rate of change of the amount of salt S in kilograms as a function of time is given by:

frac{dS}{dt} frac{S}{V}frac{dV}{dt}

Substituting the known rates:

frac{dS}{dt} frac{S}{25 2t} * 2

Further simplifying, we get:

frac{dS}{dt} frac{2S}{25 2t}

This is a separable differential equation:

intfrac{dS}{S} intfrac{2}{25 2t} dt

Integrating both sides, we obtain:

ln{S} -ln{(25 2t)} ln{c}

Solving for S, we get:

S frac{c}{25 2t}

At time t0, the initial amount of salt is 2 kg:

2 frac{c}{25}

Solving for c, we find:

c 250

Thus, the amount of salt in the tank as a function of time is:

S frac{250}{25 2t}

Evaluating this at t 15 minutes:

S frac{250}{25 2*15} frac{250}{55} approx 4.545 kg

This would, however, overestimate the amount of salt because it does not account for the continuous dilution due to the inflow of fresh water. A more accurate approach involves solving the differential equation with time-dependent volume.

The amount of salt in the tank can thus be described as:

S frac{250}{sqrt{252t^3}}

Evaluating this at t 15 minutes:

S frac{250}{sqrt{252*15^3}} approx 0.613 kg

A plot with time on the horizontal axis and amount of salt on the vertical axis would show the actual decrease in salt concentration due to continuous dilution.

Literature and Further Reading

For a deeper understanding of differential equations and their applications in engineering and science, please refer to the following resources:

1. Boyce, W. E., DiPrima, R. C. (2017). Elementary Differential Equations and Boundary Value Problems. John Wiley Sons.

2. Kreyszig, E. (2011). Advanced Engineering Mathematics. John Wiley Sons.

3. Lindfield, G. R., Penny, J. E. T. (2000). Numerical Methods Using MATLAB. John Wiley Sons.

Keywords: separation of variables, differential equations, tank mixing problem