Question

A brine solution of salt flows at a constant rate of

7

​L/min into a large tank that initially held

100

L of brine solution in which was dissolved

0.5

kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is

0.05

​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach

0.02

​kg/L?

Determine the mass of salt in the tank after t min.

massequals=nothing

kg

Answer 1

mass of salt in the tank after t min

x(t) = x kg

Initial mass of salt (t = 0)

x(0) = 0.5 kg

Salt balance

Accumulation = Input - output

dx /dt = 7 *(0.05) − 7 *(x /100)

Integrate the equation

dx/(0.35 - 0.07x) = dt

- (1 /0.07) * ln |0.35 − 0.07x| = t + C

0.35 − 0.07x = A exp (-0.07t)

x(t) = (0.35/0.07) + (A exp (-0.07t))

= 5 + (A exp (-0.07t))

Now apply the x(0) = 0.5

0.5 = 5 + A exp (-0.07*0)

A = - 4.5

Mass of salt after t minutes

x(t) = 5 - 4.5* exp (-0.07t)

When will the concentration of salt in the tank reach

x(t) = 0.02*100

5 - 4.5* exp (-0.07t) = 0.02*100

exp (-0.07t) = (5 - 0.02*100)/(4.5) = 0.666

0.07 t = 0.4064

t = 5.81 minutes