Tank Problems Worksheet

Tank problems are examples of modeling with first order linear differential equations. Let $n(t)$ represent the amount of solute, usually salt, dissolved in a total volume of $v(t)$ of solvent, usually water. The volume $v(t)$ is contained in a well mixed tank, that allows for fluid to enter and exit the tank, usually via separate pipes. Fluid entering the tank is described by the rate at which it enters the tank and its concentration. The outflow is described by the rate at which it is leaving. Its concentration is$,$ by default, given by $c_{\text{o}\text{u}\text{t}}=\frac{n}{v}.$

$\frac{dn}{dt}=r_{in}{c}_{in}-r_{\text{o}\text{u}\text{t}}\frac{n}{v}$Longleftrightarrow$\frac{dn}{dt}+(\frac{r_{\text{o}\text{u}\text{t}}}{v})n=r_{in}{c}_{in}$

See for example exercise $1\mathrm{.}4\mathrm{.}2$ in the book by Lebl for a derivation. This book is mentioned in the syllabus and you can find it easily online. The method of solution is to use an integrating factor, as you learned last week. Practice setting up and solving tank problems with the following four exercises.

Problem $1$: A well stirred tank originally contains $100$ gal of fresh water. Then brine containing $\frac{1}{4}$ lb of salt per gallon is poured into the tank at a rate of $2ga\frac{l}{m}in,$ and the mixture is allowed to leave the tank at the same rate. What is the amount of salt, in lbs$,$ in the tank at any instant?

Problem $2$: A well stirred tank with a capacity of $500$ gal originally contains $200$ gal of water with $100$lbs of salt in solution. Brine containing $1$lb of salt per gallon is pumped into the tank at a rate of $3ga\frac{l}{m}in,$ while fluid is allowed to flow out of the tank at a rate of $2ga\frac{l}{m}in.$ What is the salt concentration at the instant that the tank overflows? What about ten minutes later?

Problem $3$: A well mixed tank with a capacity of $1500$ gals originally contains $1000$ gals of fresh water. One pipe containing $\frac{1}{2}lb$ of salt per gallon is entering at a rate of $4ga\frac{l}{m}in.$ The second pipe containing $\frac{1}{3}lb$ of salt per gallon is entering at a rate of $6ga\frac{l}{m}in.$ The mixture is allowed to flow out of the tank at a rate of $5ga\frac{l}{m}in.$ Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow.

Problem $4$: A large, well stirred tank with $500$ gallons of beer contains $4\%$ alcohol $($by volume$).$ Beer with $6\%$ alcohol is pumped into the tank at a rate of $5ga\frac{l}{m}in$ and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour? How long will it take for the tank to contain $6\%$ alcohol? Sketch your solution. What does $6\%$ represent, relative to the solution, in mathematical terms?