Mixing problems occur quite frequently in chemical industry. We explain here how to solve the basic model involving a single tank (see the figure on the right). The tank contains of water in which initially of salt is dissolved. Brine – i.e. salt water – runs in at a rate of , and each gallon contains of dissolved salt. The mixture in the tank is kept uniform by stirring. Brine runs out at .
Find the amount of salt in the tank at any time
Let denote the amount of salt in the tank at time . Its time rate of change is
(1) |
times gives an inflow of of salt. Now, the outflow is of brine. This is (=1%) of the total brine content in the tank, hence 0.01 of the salt content , that is, . Thus, from (1) we obtain the following ODE as a model:
(2) |
The ODE (2) is separable. Separation, integration, and taking exponents on both sides gives
Initially, the tank contains of salt. Hence is the initial condition that will give the unique solution. Substituting and in the last equation gives . Hence . Hence the amount of salt in the tank at time is
(3) |
This function (see the graph on the right) shows an exponential approach to the limit . Can you explain physically that should increase with time? That its limit is ? Can you see the limit directly from the ODE?