Question

Consider a tank used in certain hydrodynamic experiments. after one experiment the tank contains 900 l of a dye solution with a concentration of 1 g/l. to prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 9 l/min, the well-stirred solution flowing out at the same rate. find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value. (round your answer to one decimal place.)

Answer 1

Soln: •t: in min •Q(t): quantity of dye in the tank at time t •r in= 0: rate of dye poured into the tank per min. Since it is fresh water, so it is zero. •r out= density of dye×rate of fluid =Q/900×9: rate of dye flowing out. •Q(0) = 1×900 g: initial amount of dye in the tank. •{dQ/dt=rin-rout= 0-Q100 Q(0) = 900g•Q(t)=Ce-1/100t •Q(0) = 900, C= 900, Q(t) = 900e-1/100t •Q(t) = 0.01·Q(0), Q(0)e-1/100t= 0.01×Q(0),-t100= ln 0.01, t= 460min