Fluid Mechanics Help?
Freshwater flows steadily into an open 55-gallon drum initially filled with seawater. The freshwater mixes thoroughly with the sea water and the mixture overflows out of the drum. If the freshwater flowrate is 10 gal/min, estimate the time in seconds required to decrease the difference between the density of the mixture and the density of the freshwater by 50%.
- LawrenceLv 61 decade agoFavorite Answer
This is really not a fluid mechanics problem but rather a simple first order D.E. problem...
Q(t) = the quantity of salt in tank at time t
Q(0) = the quantity of salt in tank at time zero
The salt concentration is C = Q(t) / Volume(t)
dQ/dt is the instantaneous rate of change of the salt concentration, thus
dQ/dt = (Cin)(Flow in) - (Cout)(Flow out)
dQ/dt = (Cin)(Flow in) - (Qout(t)/Volume(t)) (Flow out)
Since freshwater has zero salt, Cin = 0 and the first term drops away
For simplicity, let's let Q equal the salt in tank solution at time t
The volume in the tank is fixed at 55 gallons and the flow in, 10 GPM, equals the flow out.
dQ/dt = - ( Q lbs / 55 gal )( 10 gal/min )
dQ/dt = -10Q / 55
Move Q to the LHS and then integrate
∫ 1/Q dQ = -10/55 ∫ dt
ln Q = -10t/55 + C
Q(t) = [ e^C ] [ e^(-10t/55) ]
At t = 0 then Q(0) = e^C
So we can say...
Q(t) = Q(0)e^(-10t/55)
Let's let Q(0) = 1 and then we need to find Q(t) = 1/2
1/2 = e^(-10t/55)
t = (-55/10)ln(1/2)
t = 3.8 minutes
- itsbob1Lv 51 decade ago
Complex yet you can do this. Homework is to make you think. Get busy.