Answer to Question #236905 in Chemical Engineering for Sen. Adeleke Olumi

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
1
Expert's answer
2021-09-15T02:13:42-0400

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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