Question #236905

A tank contains 10m^3 of free water, brine having a concentration of 15kgsalt/m^3 is sent into the tank at the rate of 200litres/min. The mixture is kept uniform by mixing and runs out at the rate of 100litres/min. What will be the exit brine concentration when the tank contains 15m^3 of brine

Expert's answer

initial amount of salt = "10m^3"

rate of pumping in = 200L/min

Let A be amount of salt at any time t,

then

"\\frac{dA}{dt} = R_{in} - R_{out}"

R is rate of flow.

"R_{in} = \\frac{1g}{1L} \\frac{5L}{1 min} = 5 g\/min"

"R_{out} = \\frac{A}{200}*{5} = \\frac{A}{40} g\/min"

Hence equation of state will be,

"\\frac{dA}{dt} = 5-\\frac{A}{40}"

solving equation,

"\\frac{dA}{dt} + \\frac{1}{40}A =5"

"e^{\\frac{1}{40}t}A = \\int 5e^{\\frac{1}{40}t}dt"

"e^{\\frac{1}{40}t}A = 200e^{\\frac{1}{40}t} + C"

"A = 200 + Ce^{-\\frac{1}{40}t}"

applying condition, A=20 at t=0

"20 = 200 + C \\implies C = -180"

Hence desired equation of the state is given by,

"A = 200 - 180e^{-\\frac{1}{40}t}"

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