A tank originally contains 320 gal of fresh water Then water

A tank originally contains 320 gal of fresh water. Then water containing a half lb of salt per gal is poured into the tank at a rate of 8 gal/min, and the mixture is allowed to leave at the same rate. After 15 minutes the salt water solution flowing into the tank suddenly switches to fresh water flowing in at a rate of 8 gal/min while the solution continues to leave the tank at the same rate. Find the Laplace transform of the amount of salt y(t) in the tank.

Solution

t: in min

• Q(t): the amount of salt in the tank at time t.

• rin = 1/2 × 8= 4 lb/min: rate of salt poured into the tank 1

• rout = Q/320× 8 = Q/40 lb/min: rate of salt flowing out from the tank

• Q(0) = 0: initial condition, because of fresh water at the beginning.

• dQ/dt = rin rout = 1 Q /40

Q(0) = 0

• Facts: If y\' = ay + b , general solution: y = Ce^at b/a

• Q(t) = Ce^-t/40 + 40

• Q(0) = C +40 = 0, Q(t) = 40e^-t/160 + 40.

• Q(15) = 40e^-15/40 + 40 = 12.51

Second 15 mins:

• rin = 0 × 8 = 0 lb/min

• rout = Q/320 × 8 lb/min

• ( dQ/dt = rin rout = 0 Q/40,

Q(0) =12.51

• Q = Ce^-t/40

• Q(0) = C = 12.51, Q(t) = 12.51e^-t/40

•let Laplce transform of Q(t) =W(s)

W(s)=12.51/(s+(1/40))