Chapter 2.3, Problem 35E

Mixing Suppose a brine containing $0.2\text{}\text{kg}$ of salt per liter runs into a tank initially filled with $500\text{L}$ of water containing $5\text{kg}$ of salt. The brine enters the tank at a rate of $5\text{L}/\text{min}$. The mixture, kept uniform by stirring, is flowing out at the rate of $5\text{L}/\text{min}$ (see Figure 2.6).

Figure 2.6 Mixing problem with equal flow rates

a. Find the concentration, in kilograms per liter, of salt in the tank after $10\min$.[Hint: Let $A$ denote the number of kilograms of salt in the tank at $t$ minutes after the process begins and use the fact that the rate of increase in $A=\text{rate of input-rate of exit}$.

A further discussion of mixing problems is given in Section 3.2.]

b. After $10\text{min}$, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7).What will be the concentration, in kilograms per liter, of salt in the tank $20\min$ after the leak develops? [Hint: Use the method discussed in Problems 31 and 32.]

Figure 2.7 Mixing problem with unequal flow rates