A tank contains 20 litres of pure water. A stream of brine with a concentration of 3 gms of salt per 100 ml of solution is fed into the tank at the rate of 0.5 lit/sec, and another... A tank contains 20 litres of pure water. A stream of brine with a concentration of 3 gms of salt per 100 ml of solution is fed into the tank at the rate of 0.5 lit/sec, and another stream of sugar solution (concentration of 10 gms per 100 ml) is fed at the rate of 1 lit/sec. If the tank is well agitated, what are the concentrations of salt and sugar in the tank when it contains 29 litres of solution?

Understand the Problem

The question asks for the concentrations of salt and sugar in a tank when it contains 29 liters of solution. It provides details on the initial volume, the flow rates, and concentrations of the incoming brine and sugar solutions. To solve this, we would need to set up a mass balance for salt and sugar, accounting for the volumes and rates at which they enter the tank.

Answer

Salt concentration: $3.1 \text{ g/L}$, Sugar concentration: $20.7 \text{ g/L}$.

Steps to Solve

Initial Setup of Data The tank starts with 20 liters of pure water. We need to calculate the concentrations when the tank has 29 liters of solution. The brine has a concentration of 3 g/100 ml and enters at a rate of 0.5 liters/sec. The sugar solution has a concentration of 10 g/100 ml and enters at the rate of 1 liter/sec.
Convert Concentrations to g/L Convert the concentrations of salt and sugar from g/100 ml to g/L for easier calculations:

Salt concentration: ( 3 \text{ g/100 ml} = 3 \times 10 = 30 \text{ g/L} )
Sugar concentration: ( 10 \text{ g/100 ml} = 10 \times 10 = 100 \text{ g/L} )

Calculate the Volume of Solutions Added The flow rates are given in liters/second. To find how long it takes to reach a total of 29 liters from the initial 20 liters, we need 9 more liters.

At a total inflow rate of ( 0.5 \text{ L/sec} + 1 \text{ L/sec} = 1.5 \text{ L/sec} ), the time ( t ) required is $$ t = \frac{9 \text{ L}}{1.5 \text{ L/sec}} = 6 \text{ seconds} $$

Calculate Mass of Salt and Sugar Added Determine how much salt and sugar are added in 6 seconds:

Salt added: $$ \text{Salt} = 30 \text{ g/L} \times 0.5 \text{ L/sec} \times 6 \text{ sec} = 90 \text{ g} $$
Sugar added: $$ \text{Sugar} = 100 \text{ g/L} \times 1 \text{ L/sec} \times 6 \text{ sec} = 600 \text{ g} $$

Total Mass in Tank The total mass of each in the tank after 6 seconds:

Total mass of salt in the tank = 90 g
Total mass of sugar in the tank = 600 g

Calculate Concentration in 29 Liters To find the concentrations when the tank holds 29 liters:

Concentration of salt: $$ \text{Concentration of salt} = \frac{90 \text{ g}}{29 \text{ L}} \approx 3.1 \text{ g/L} $$
Concentration of sugar: $$ \text{Concentration of sugar} = \frac{600 \text{ g}}{29 \text{ L}} \approx 20.7 \text{ g/L} $$

The concentration of salt is approximately $3.1 \text{ g/L}$ and the concentration of sugar is approximately $20.7 \text{ g/L}$ when the tank contains 29 liters of solution.

More Information

This problem involves mass balance techniques used in chemical engineering to find concentrations from flow rates and concentrations of incoming solutions. The initial conditions of the tank change dynamically as new solutions are added.

Tips

Miscalculating the total volume added when inflow rates are combined.
Incorrectly converting concentrations from g/100 ml to g/L.
Forgetting to consider the time required to reach the target volume.

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